Measurement Uncertainty with a Smartphone

Q: Are smartphone sensors reliable?

They are reliable enough for a great many scientific experiments, especially for comparing situations and tracking changes. Their absolute accuracy, however, varies by model and is not guaranteed without calibration. The right question is not "reliable or not" but "sufficient or not for the question asked" (Part 19).

Q: Why does a measurement vary?

Because many small causes add up: electronic noise, vibrations, temperature, environment, the operator's gesture, software rounding. This variability is universal; you study it by repeating the measurement (Part 2).

Q: What do the digits after the decimal point mean?

They reflect the display format chosen by the software, not the quality of the measurement. Only the digits that stay stable at the level of the uncertainty are significant; decimals fluttering under the noise carry no information (Parts 6 and 14).

Q: What is the difference between precision and resolution?

Resolution is the smallest step the instrument can represent or display; precision describes how tightly repeated results cluster. A high-resolution instrument can be imprecise, and vice versa (Part 6).

Q: What is the difference between accuracy and precision?

Precision describes the clustering of repeated measurements (small spread); accuracy describes the absence of bias with respect to an accepted value. A series can be precise and wrong, or scattered and accurate on average (Part 4).

Q: What is the difference between error and uncertainty?

The error is the actual - generally unknown - gap between the result and the value sought. The uncertainty is the calculable estimate of the doubt attached to the result (Part 5).

Q: What is the standard deviation for?

To quantify the spread of a series of measurements around its mean. It measures neither the error nor the accuracy: two series with the same standard deviation can be one centered on the accepted value, the other far off (Part 3).

Q: Why repeat a measurement?

To make the variability visible, detect outliers, and reduce the statistical uncertainty of the mean, which shrinks like s/√n (the standard error). Repetition, however, corrects no bias (Parts 3 and 11).

Q: How many measurements should you take?

There is no magic number: about ten measurements already give a good picture of the spread, and the uncertainty of the mean then decreases only slowly (as 1/√n). Ten measurements under a stable procedure beat a hundred under changing conditions (Part 11).

Q: What is a Type A uncertainty?

A standard uncertainty evaluated statistically from a series of repeated measurements: u_A = s/√n under the right conditions - the formal name for what lab manuals call the random or statistical uncertainty (Part 11).

Accuracy, errors, calibration and confidence intervals with digital sensors and FizziQ

A smartphone lying motionless on a table can display 9.79, then 9.83, then 9.80 m/s² in succession - or sometimes a value with six decimal places. What should you make of it? This guide uses smartphone sensors and the FizziQ app to make the concepts of variability, error, bias, calibration and uncertainty concrete, from high school science through first-year college labs. Because going digital does not remove uncertainty: it only makes it harder to see.

Introduction - Is a digital readout necessarily precise?
What is a measurement?
Why do results vary?
How do you summarize a series of measurements?
Accuracy, precision, and the illusion of exactness
Error, bias and uncertainty
Resolution, sensitivity and noise
Sampling rate and fast phenomena
Measuring range and saturation
Repeatability and reproducibility
Calibrating a digital instrument
Evaluating uncertainty statistically (Type A)
Evaluating uncertainty from other information (Type B)
Combining simple uncertainties
Reporting a result correctly
Comparing a measurement with an accepted value
Measurement uncertainty and confidence intervals
The sensors as case studies
Twelve classroom activities with FizziQ
When is a smartphone good enough?
Common mistakes and confusions
Frequently asked questions
Conclusion
Sources and references

Introduction - Is a digital readout necessarily precise?

Lay a smartphone perfectly still on a table, open the Absolute acceleration instrument (or Acceleration with gravity) in FizziQ, and watch the screen. The values scroll by:

9.79 ; 9.83 ; 9.80 ; 9.82 m/s²

And depending on the display format, the app may even show a value such as:

9.813742 m/s²

Six questions immediately arise - they are the questions of this guide. Why do the values vary when the phone isn’t moving? Which value should you keep? Are all those digits significant? What is the uncertainty of the result? Is the result consistent with an accepted value of g? And would two other smartphones give the same thing?

The answer fits in one sentence: a smartphone can produce scientifically useful measurements, but the readout alone tells you nothing about their quality. You have to study the variability, the procedure, the biases, the resolution, the calibration and the uncertainty. That is exactly what students learn, in increasing depth, from introductory high school science through first-year college labs - and the smartphone is a wonderful tool for learning it, precisely because it quickly produces large amounts of real, imperfect data.

What students learn at each level. This guide deliberately covers the whole progression; each teacher can draw what fits their course.

Introductory high school science (9th-10th grade, physical science, chemistry, physics): observing and describing variability (repeated trials, histogram, mean, standard deviation), accuracy vs. precision, significant figures, and reporting a result sensibly.

Honors, AP and IB courses (11th-12th grade): estimating the uncertainty of a result from repeated trials (standard error of the mean) or from the instrument’s specifications, drawing error bars, and judging whether a result agrees with an accepted value.

AP/IB lab work and 12th grade: combining uncertainties with a supplied rule (quadrature for products and quotients), and comparing a result with a reference quantitatively, in units of the uncertainty.

First-year college: Type A and Type B evaluations in the NIST/GUM sense, and building and correctly interpreting a confidence interval (Student’s t).

With analog instruments, the limits of reading are visible: you can see the size of a ruler’s graduations, the width of the line, the position of the needle, the parallax, the difficulty of interpolating between two marks. You know instinctively how many digits to keep. With a digital instrument, the screen displays decimals with misleading confidence: the number of decimal places displayed tells you nothing, by itself, about the quality of the measurement. The quality of a digital measurement depends on the sensor, its calibration, its effective resolution, its noise, its measuring range, its sampling rate, the software processing, the procedure, the environment, the operator, and the statistical treatment of the data.

Throughout this guide, FizziQ serves as the main tool: the app gives access to the sensors, records series of measurements, draws graphs and histograms, supports statistical calculations in its built-in data table, and documents everything in an experiment notebook. All the concepts presented remain general, however: they apply to any app, to external sensors, and to analog and laboratory instruments alike.

Part 1 - What is a measurement?

1.1 A measurement is not just a number

To measure is to assign a numerical value to a quantity (a length, a duration, an acceleration, a sound level…) expressed in a unit. A few precise terms, illustrated with the accelerometer:

The measurand is the quantity you are trying to measure - for example the local acceleration due to gravity, g.
The reading (or indication) is what the device displays - for example 9.82 m/s².
The measured value is the value assigned to the measurand from one or more readings, after applying the procedure (averaging, corrections…).
The measurement result is the complete package: a value, a unit, the conditions under which the measurement was made, and an estimate of the uncertainty when relevant.
The procedure (protocol) describes how you go about it: position of the device, duration, number of trials, experimental conditions.

A number by itself - “9.82” - is not a measurement result. “g = (9.81 ± 0.06) m/s², smartphone lying on a stable table, average over a 30 s recording” is one.

1.2 What does the smartphone actually measure?

Before interpreting a number, you must understand how that number was produced. And smartphone sensors do not always measure what their names suggest:

The accelerometer does not measure “the acceleration of the motion” in the physics-class sense: it measures a specific force that includes the effect of gravity. That is why a motionless phone reads about 9.81 m/s² and not zero. FizziQ actually offers several distinct instruments - linear acceleration (gravity removed), acceleration with gravity, absolute acceleration - which are different software treatments of the same sensor.
The magnetometer measures three components of the magnetic field; the displayed “magnitude” is computed from those components.
The sound level meter does not measure acoustic pressure directly: it displays a level in decibels computed from the microphone’s digitized signal, with an approximate calibration.
GPS does not strictly measure an instantaneous speed: speed is determined from the signal and successive positions.
Video analysis does not measure a speed: it records clicked positions, from which speeds and accelerations are computed by differences.

1.3 Direct measurement and computed quantity

You therefore need to distinguish four levels, often hidden behind the same display:

the value coming directly from a sensor (the raw components of the accelerometer);
the value obtained after filtering (FizziQ, for instance, offers an adjustable smoothing of the signal in its settings);
the value obtained by sensor fusion (the phone’s orientation combines accelerometer, gyroscope and magnetometer);
the value computed from other measurements (a speed v = d/t, an acceleration derived from clicked positions).

Every computational step transforms the measurement - and transforms its uncertainty too, as we will see in Part 13.

Key point. Before interpreting a displayed number, ask: what physical quantity does the sensor actually detect, and what computations separate that detection from the number on the screen?

Part 2 - Why do results vary?

2.1 Repeating a measurement

The founding experiment of this whole guide fits in one instruction:

Lay a smartphone motionless on a table and record the acceleration for thirty seconds with FizziQ.

The result always surprises: the curve is not a perfectly flat line. The values fluctuate around a central value. The same operation, repeated under apparently identical conditions, does not always give exactly the same result - this is the variability of measurement, and it is universal: it exists with laboratory instruments too, just sometimes at smaller scales.

2.2 Sources of variability

Where do these fluctuations come from? Many sources add up: the sensor’s own electronic noise; vibrations from the table, the building, footsteps in the hallway; instability of the device’s position; temperature variations that slightly change the sensor’s response; the electromagnetic environment (especially for the magnetometer); fluctuations of the source being measured; the operator’s gesture and the triggering (no two stopwatch starts are ever identical); the software’s filtering algorithms and rounding; and genuine changes over time in the phenomenon being studied.

Some of these causes produce random fluctuations; others, as we will see, produce systematic offsets. Learning to tell them apart is one of the central goals of this guide.

2.3 Displaying a series

A series of measurements should be described graphically first, before any calculation:

the data table (FizziQ automatically builds a table when you take several instant captures with the same instrument);
the time graph, which shows the fluctuation over time and reveals any drift;
the histogram, which shows how the values are distributed (in FizziQ, the “Statistics” display mode of a graph shows the counts of the different measured values; in FizziQ Web and in Python, you plot the histogram from the exported data);
the minimum, the maximum, and the range R = x_max − x_min, a first numerical description of the spread.

2.4 Outliers

Sometimes a value appears that lies far from the others. Before discarding it, look for its cause: it may come from a real disturbance (a bump on the table), a handling mistake, a data loss, a one-off malfunction - or from a genuine physical phenomenon that it would be wrong to eliminate without justification.

Key point. Never delete a value automatically just because it is different from the others. Delete it only if you can identify and document the mistake or disturbance that explains it.

Part 3 - How do you summarize a series of measurements?

3.1 The mean

The arithmetic mean of a series of n measurements x₁, x₂, …, xₙ is:

x̄ = (1/n) Σᵢ₌₁ⁿ xᵢ

where x̄ is the mean, n the number of measurements and xᵢ the i-th measured value. The mean provides a central value for the series - it is usually what you keep as the measured value. But it says nothing about the spread: two series with the same mean can be one tightly clustered, the other widely scattered.

Example (the series from the introduction): x̄ = (9.79 + 9.83 + 9.80 + 9.82)/4 = 9.810 m/s².

3.2 The range

The range is the simplest first description of spread:

R = x_max − x_min

It is immediate to compute and understand, but fragile: it depends only on the two extreme values, so a single outlier makes it explode, and it mechanically grows with the number of measurements.

3.3 The sample standard deviation

The sample standard deviation characterizes the spread of the observations around the mean:

s = √( Σᵢ₌₁ⁿ (xᵢ − x̄)² / (n − 1) )

where s is the sample standard deviation, xᵢ the measured values, x̄ their mean and n their number. The larger s is, the more scattered the measurements. The standard deviation has the same unit as the measured quantity.

Example: for the series 9.79, 9.83, 9.80, 9.82 m/s², the deviations from the mean are −0.02, +0.02, −0.01, +0.01, so s = √((0.0004 + 0.0004 + 0.0001 + 0.0001)/3) ≈ 0.018 m/s².

A word of caution about vocabulary: the standard deviation is not “the error” of the measurement. It describes the spread of the observations, nothing more.

3.4 Spread versus uncertainty of the mean

This is the most commonly misunderstood distinction in the whole chapter:

s describes the spread of the individual measurements: one more measurement will typically land within ± s of the mean. Taking more measurements does not shrink s: the spread is a property of the measurement process.
s/√n - called the standard error of the mean - characterizes, under the right conditions (see Part 11), the statistical uncertainty of the mean: the more measurements you accumulate, the better the mean is pinned down, even though each individual measurement remains just as scattered as before.

In other words: repeating measurements does not improve each measurement, but it improves your knowledge of their average.

3.5 Where this sits in US courses

Histograms, mean and standard deviation for repeated trials appear in introductory high school science (and in parallel in statistics classes); the finer interpretation of s/√n as the standard error of the mean is typically developed in honors, AP and IB lab work and consolidated in first-year college laboratories.

Part 4 - Accuracy, precision, and the illusion of exactness

4.1 Precision

A series of measurements is precise when the repeated results are tightly clustered: the values resemble one another. Precision is read from the standard deviation: small s, high precision.

4.2 Accuracy

A measurement method is accurate when its bias with respect to an accepted value is small: the average of a large number of measurements falls close to the accepted value. Accuracy cannot be seen from the series alone - you need an external reference to assess it. (In formal metrology, this “low bias” property is called trueness, and “accuracy” denotes the overall quality; in the classroom, the everyday accuracy/precision pair works fine as long as both ideas stay distinct.)

4.3 Four situations

These two qualities are independent, which yields four possible situations:

Situation	Spread	Distance of the mean from the accepted value
Accurate and precise	Small	Small
Precise but biased	Small	Large
Accurate on average but imprecise	Large	Small
Neither accurate nor precise	Large	Large

The classic dartboard image (arrows clustered or not, centered or not) illustrates the four cases well - but it is only an illustration: in reality you cannot “see” the center of the target, because the true value is unknown.

4.4 Exactness and the trap of the display

A common language trap: in everyday speech, “precise” often means “with many decimal places.” In measurement science it does not. A device that displays many digits is not thereby precise (its readings may scatter widely), not thereby accurate (its mean may be far off), and not thereby calibrated. The display is a formatting choice, not a metrological quality. Keep three ideas separate: precision (small spread), accuracy (small bias), and resolution (small display step, Part 6) - a device can have any combination of the three.

4.5 An example with FizziQ

Have g measured (acceleration with gravity, phone lying flat, 30 s recording, average) by two different smartphones:

the first gives tightly clustered values (small s) but with a mean of 9.95 m/s²;
the second gives more scattered values, but with a mean of 9.81 m/s².

Which device is more precise? The first. Which seems more accurate? The second. Which result is more useful? It depends on the question: to track a change in acceleration, the first device’s precision is precious; to determine the value of g, the second’s accuracy wins - and the first would become excellent after calibration (Part 10).

Part 5 - Error, bias and uncertainty

This part is the conceptual core of the guide.

5.1 Measurement error

The measurement error is the difference between a measured value and the true value of the measurand (or an accepted reference value standing in for it). The fundamental difficulty: the true value is generally unknown - if you knew it, you would not be measuring. The actual error therefore generally cannot be computed exactly. What you can do is study its components and estimate its likely size.

5.2 Bias, or systematic error

A bias (systematic error) is a component of the error that repeats identically from one measurement to the next. Concrete smartphone examples: a mis-set accelerometer zero; an incorrect scale factor (the sensor “counts” 2% too much); a misalignment (the phone is not exactly level); a mismeasured reference distance in video analysis; an uncalibrated microphone that shifts every sound level; the permanent influence of a magnet or a steel structure on the magnetometer; a perspective error in video tracking; software processing (filtering, automatic gain) that systematically distorts the signal.

Essential property: a bias does not go away when you repeat the measurement. A thousand measurements with a shifted zero give a very well determined average… of the wrong result.

5.3 Random scatter

Random scatter groups the unpredictable variations around the average value: noise, micro-vibrations, small differences in gesture. Unlike bias, its effects partially cancel out: repeating and averaging reduces the statistical uncertainty that flows from it (that is the s/√n of Part 3).

5.4 The difference between error and uncertainty

Box - Error vs. uncertainty. The error is the actual gap, generally unknown, between the result and the value sought. The uncertainty characterizes the doubt attached to the result: a calculable estimate of the size of the plausible errors. You do not know your error; you evaluate your uncertainty.

Do not say that uncertainty is “the probable error” without explanation: uncertainty does not designate any particular error, but the width of the reasonable doubt around the result, given everything you know about the measurement process.

5.5 Blunders

A blunder is neither random error nor a bias to fold into the uncertainty: it is an identifiable failure of the procedure. Examples: using the wrong unit; recording the Y axis instead of the Z axis; a data-entry mistake; covering the sensor with a finger; moving the device during the reference measurement; using the wrong formula.

Key point. An identified blunder gets corrected (you redo the measurement or the calculation); it is never artificially folded into an uncertainty.

Part 6 - Resolution, sensitivity and noise

6.1 Display resolution

The display resolution is the smallest step visible on screen: if the app shows 9.8137, the display step is 0.0001. That step often results from a mere formatting choice in the software - it says nothing about the sensor’s quality. In FizziQ’s data tables, the number of displayed decimal places is in fact a setting the user can adjust column by column.

6.2 Nominal sensor resolution

The nominal resolution is the smallest variation the hardware chain (sensor + analog-to-digital converter) can theoretically represent: it depends on the component and the number of bits of the conversion. It varies from one smartphone model to another and is generally not documented for the end user.

6.3 Effective resolution

The effective resolution is what you can actually distinguish in practice - and it is often limited by noise, not by the digital step. Typical example: the display shows thousandths, but the signal of a motionless sensor fluctuates by several hundredths. The displayed thousandths then change constantly and randomly: they are not significant. The noise, visible by recording the sensor at rest, sets the practical limit of the measurement.

Box - Resolution vs. precision. Resolution is the smallest step the instrument can display or represent; it guarantees neither precision nor accuracy. An instrument can display thousandths (high resolution) while fluctuating by several hundredths (effective resolution limited by noise) and while being offset by several tenths (bias). The quality of a measurement cannot be read from the number of decimal places.

6.4 Sensitivity

The sensitivity of a sensor is the change of its output for a given change of the input quantity. A very sensitive sensor detects small variations - which makes it neither accurate nor precise for all that: sensitivity, resolution and accuracy are three distinct notions.

6.5 Activity with FizziQ

Record 30 seconds of a motionless sensor (accelerometer or magnetometer), then answer: how many decimal places are displayed? What is the amplitude of the fluctuations (the range, read by zooming into the graph)? Which decimal place actually looks stable? How should the result be rounded? This twenty-minute activity is enough to establish the guiding idea of the guide: displayed decimals and significant decimals are two different things.

Part 7 - Sampling rate and fast phenomena

7.1 Definition

The sampling rate is the number of values acquired per second, in hertz. Several rates must in fact be distinguished, and they do not necessarily coincide: the rate at which the sensor produces data; the acquisition rate the app retains (adjustable in FizziQ, within the sensor’s limit); the refresh rate of the display; and the rate of the exported data. In FizziQ, the maximum rates depend on the sensor and the device model: typically a few dozen to about a hundred hertz for the accelerometer or gyroscope, about one hertz for GPS or the barometer, and much more for the microphone.

7.2 Consequences of undersampling

A system that samples too slowly can: miss a maximum (the peak of an impact falls between two samples); underestimate an impact for the same reason; distort a fast motion, which looks angular or smoothed; give a wrong frequency for an oscillation; or even create a false slow signal that does not exist in the phenomenon (aliasing).

Going further - first-year college. The Shannon-Nyquist criterion states that a signal can only be represented correctly if the sampling rate exceeds twice the highest frequency present in the signal. Below that, fast components “fold back” into false slow components. This principle, stated here without proof, explains the false signals you see when filming a wheel that seems to spin backwards.

7.3 Activity

With the same acquisition setting, record in succession: a slow motion (lift and gently set down the phone), a fast vibration (phone resting on a speaker), a brief impact (a light tap on the table). Compare the three recordings: which information is correctly captured, and which is obviously truncated or distorted? Repeat the experiment after lowering the sampling rate in FizziQ’s settings to make the degradation visible.

Part 8 - Measuring range and saturation

8.1 Measuring range

The measuring range is the interval of values within which the sensor can deliver a usable reading. Below it, the signal drowns in noise; above it, the sensor saturates. Every smartphone sensor has its range, which varies by model and is not always documented.

8.2 Saturation

Saturation shows characteristic signs: the value plateaus; the same maximum value repeats exactly; the peaks of an oscillating signal are flattened; the display stops rising while the phenomenon obviously intensifies; the signal is distorted (in audio, this is called clipping).

8.3 Stable but wrong

A crucial point: a saturated measurement can be extremely stable - the sensor imperturbably displays its maximum value. The absence of fluctuation therefore does not guarantee the quality of a measurement: a saturated signal, or one heavily filtered by software, can be both very stable and completely wrong. Stability is a pleasant condition, not a proof.

8.4 Activity

Produce a phenomenon of increasing intensity - shake the phone harder and harder for the accelerometer, or gradually bring a loud sound source closer to the microphone - and spot the moment when the display stops evolving normally: curve plateauing, flattened peaks. Note the plateau value observed: it is an experimental estimate of the range limit of your own device.

Part 9 - Repeatability and reproducibility

9.1 Repeatability

Repeatability characterizes the spread obtained when the measurement is repeated under conditions as identical as possible: same device, same operator, same procedure, same conditions, over a short time. It is the “floor” spread of the measurement process.

9.2 Reproducibility

Reproducibility characterizes the spread obtained when a significant element changes: the device, the lab group, the day, the room, the operating-system version, the phone’s orientation, or the method itself. Reproducibility is almost always worse than repeatability - and the gap between the two is rich in lessons.

9.3 A whole-class experiment

Have the same phenomenon measured - for instance g at rest, or the sound level of a fixed source - by every smartphone in the class, each one repeating the measurement several times. The analysis then separates four things: the spread of the repetitions on each device (individual repeatability); the differences between the devices’ means (instrument effect); the operator effects (gestures, positioning); and the environment effects (place in the room, vibrating table). FizziQ’s data table, or pooling the observations via QR codes, makes it easy to gather and compare these series.

9.4 The teaching opportunity

The diversity of smartphones in a class is often experienced as a defect. It is actually a rare teaching opportunity: a fleet of instruments that are all different, which makes it possible to teach reproducibility, instrument comparison, the need for a shared procedure, and the limits of any generalization concretely (“the smartphone measures g to 1%” is meaningless; “this phone, that day, with this procedure” means something).

Part 10 - Calibrating a digital instrument

10.1 An accessible definition

Calibrating an instrument means comparing its readings with reference values in order to characterize the relationship between the two. Calibration answers the question: when the instrument displays x, what is the corresponding reference value?

10.2 Calibration and zeroing

Several operations of increasing complexity are often lumped together under the word “calibration”: zeroing (also called taring - making the device read zero in the absence of the phenomenon); offset correction (subtracting a constant bias); scale-factor correction (multiplying by a coefficient); multi-point calibration (characterizing the relationship over the whole useful range). Zeroing only fixes an offset: it guarantees nothing about the rest of the range.

10.3 Adjustment and correction

Two strategies exist once the discrepancy is characterized: adjust the instrument (change its setting so that it reads correctly - this is what FizziQ’s sound-level-meter calibration setting does, applying an offset in decibels, or the magnetometer calibration by figure-eight rotations); or apply a correction to the results afterwards, in the data table. In both cases, two good practices: keep the raw data, and state the correction method in the lab report.

10.4 Domain of validity

A correction determined in one situation is not automatically valid everywhere: it may not apply to all values (a sound level meter corrected at 60 dB may still be wrong at 90 dB), to all frequencies, to all orientations of the sensor, or to all environments. A calibration has a domain of validity, which must be made explicit.

10.5 Case study: the accelerometer and gravity

Gravity provides a natural, free reference. Procedure: place the phone successively in three orientations - the axis under study pointing vertically up, vertically down, then horizontal - and record a few seconds in each position. You then study: the zero (the reading in the horizontal position, which should be null); the readings near +g and −g; the spread in each position; a possible bias (for example, if the half-sum of the up/down readings is not zero, the zero is offset; if the half-difference departs from g, the scale factor is off); and a simple correction (offset and factor) that can be applied in the data table. This is Activity 5 in Part 18.

10.6 Case study: the sound level meter

Compare the FizziQ sound level meter with a reference meter, rigorously keeping: same position, same orientation, same source, same frequency, same measurement duration. The observed discrepancy provides the correction to apply - in FizziQ, directly via the sound-level-meter calibration setting. But mind the domain of validity: a single decibel correction, determined at one frequency and one level, does not guarantee accuracy at all frequencies and levels, because the microphone’s response is not uniform.

Part 11 - Evaluating uncertainty statistically (Type A)

11.1 Principle

A Type A evaluation of uncertainty rests on the statistical analysis of a series of repeated measurements: you let the data themselves reveal the spread of the process. (The Type A / Type B vocabulary comes from the international uncertainty guide used by NIST; in high school labs the same idea usually goes by “random/statistical uncertainty.”)

11.2 Mean and standard deviation

The tools are those of Part 3:

x̄ = (1/n) Σᵢ₌₁ⁿ xᵢ and s = √( Σᵢ₌₁ⁿ (xᵢ − x̄)² / (n − 1) )

In FizziQ, the mean and standard-deviation functions of the built-in data table perform these calculations on a column of measurements; as with any spreadsheet, check that the standard-deviation function used is the sample standard deviation (denominator n − 1). The CSV or Python export allows the same treatment in a regular spreadsheet or in code.

11.3 Standard uncertainty of the mean

Under the right conditions, the standard uncertainty associated with the mean is the standard error of the mean:

u_A(x̄) = s/√n

This expression assumes, in particular: genuinely repeated observations; sufficiently stable conditions during the series; scatter of random character; no significant drift (no rising temperature, no battery effect); and values that are sufficiently independent of one another (a heavily filtered signal produces correlated successive values, which makes s/√n too optimistic).

Complete example: the series 9.79, 9.83, 9.80, 9.82 m/s² gives x̄ = 9.810 m/s², s ≈ 0.018 m/s², and u_A(x̄) = 0.018/√4 = 0.009 m/s².

Box - Standard deviation vs. standard error. The standard deviation s describes the spread of individual measurements: it does not shrink as you take more of them. The standard error of the mean u_A = s/√n describes the doubt on the average value: it shrinks as n grows. Confusing the two leads to over- or under-estimating the uncertainty by a factor of √n.

11.4 What Type A does not fix

Statistical evaluation only sees the spread. It neither removes nor detects: a bad calibration; a zero error; a formula error; saturation; or any bias common to all the measurements. A tiny Type A uncertainty can accompany a very wrong result.

11.5 Activity

Measure the same quantity 5, then 10, then 30 times (FizziQ’s instant captures, automatically gathered into a table, are well suited). Compare, for each sample size: the mean, the standard deviation, and the standard error of the mean. Observe that s stays roughly stable while s/√n decreases - and explain why: the spread is a property of the process, the doubt on the mean depends on the statistical effort.

Part 12 - Evaluating uncertainty from other information (Type B)

12.1 Principle

A Type B evaluation uses information other than a statistical series: the resolution of a display, a manufacturer’s specification, an instrument’s tolerance, the uncertainty of a reference value, the uncertainty of a distance measured with a tape, a reading interval, the estimated effect of orientation, or the frame rate of a video. It is used in particular when only a single measurement is available. (In US high school labs, the corresponding rule of thumb is ”± half the smallest division”; Type B is the general framework behind it.)

12.2 The uniform-interval case

If you can assert that the value lies somewhere between x − a and x + a, with no reason to favor any part of that interval (assumed uniform distribution), you adopt as the standard uncertainty:

u_B(x) = a/√3

where a is the half-width of the interval. Two precautions: the uniformity assumption must be justified (reasonable for a display rounding, debatable for other situations); and this formula does not automatically apply to every “resolution” - it assumes the resolution really is the dominant source of doubt, which is false if the noise is larger than the display step (Part 6).

12.3 Example with a ruler

For a ruler graduated in millimeters, a position is read to about half a division: a = 0.5 mm, so u_B ≈ 0.5/√3 ≈ 0.3 mm. The teaching value of the analog example is to make visible what digital hides: you can see the graduation that limits the reading.

12.4 Digital examples

The same logic applies to digital instruments:

Video frame rate at 30 fps: an event (release, impact) is dated to half a frame, a = 1/60 s, so u_B = 1/(60√3) ≈ 0.010 s.
Stopwatch step of one hundredth: a = 0.005 s, u_B ≈ 0.003 s.
Resolution of a distance read from a map or plan at 1 m steps: a = 0.5 m, u_B ≈ 0.3 m.
A sensor’s display step, only if the signal is more stable than that step - otherwise the noise dominates and a Type A evaluation is required.

Part 13 - Combining simple uncertainties

This part mainly concerns AP/IB lab work and first-year college.

13.1 Several contributions

A real measurement rarely depends on a single source of doubt. Measuring a speed, for instance, accumulates: the repeatability of the timing (Type A), the time resolution (Type B), any calibration, the uncertainty on the distance, the one on the duration.

13.2 Adding in quadrature

For independent contributions, each already expressed as a standard uncertainty in the same unit, the combined standard uncertainty is:

u_c = √(u₁² + u₂² + …)

This “root-sum-square” addition assumes the sources are independent: two contributions arising from the same cause do not combine this way. A useful practical consequence: the largest contribution quickly dominates - if u₁ = 3 u₂, removing u₂ entirely would change almost nothing. Uncertainty analysis serves first to identify the dominant source, the one worth working on.

13.3 Computed quantity

For a computed quantity, high school labs use a supplied rule suited to the case at hand. The standard example, speed v = d/t:

u(v)/v = √( (u(d)/d)² + (u(t)/t)² )

Worked example: d = (2.00 ± 0.01) m and t = (0.58 ± 0.02) s give v = 3.45 m/s, with u(d)/d = 0.5% and u(t)/t = 3.45%, so u(v)/v = √(0.005² + 0.0345²) ≈ 3.5% and u(v) ≈ 0.12 m/s: v = (3.45 ± 0.12) m/s. Note in passing that the duration utterly dominates the uncertainty - it is the one worth measuring better.

Going further - first-year college. The formula above is the special case of uncertainty propagation for products and quotients: relative uncertainties add in quadrature. The general case, based on the partial derivatives of the function relating the quantities, is studied in first-year college; it can also be explored numerically by simulation, drawing the input quantities at random within their uncertainty intervals.

13.4 Relative uncertainty

The relative uncertainty (also called fractional or percent uncertainty) relates the uncertainty to the measured value:

u_r(x) = u(x)/|x|

often expressed as a percentage. It makes it possible to compare the quality of measurements of very different magnitudes: an uncertainty of 1 mm is excellent on a 10 m distance (0.01%) and poor on a 5 mm thickness (20%).

Part 14 - Reporting a result correctly

14.1 Unit

A value without a unit is generally incomplete: “9.81” means nothing, “9.81 m/s²” is a measurement. The unit is part of the result, in the text as well as in tables and on graph axes (FizziQ’s tables let you attach the unit to each column name).

14.2 Significant figures

Do not automatically copy every displayed decimal: keep the digits the measurement justifies, that is, those that remain stable at the level of the uncertainty. Decimals that fluctuate freely under the noise (Part 6) are not significant.

14.3 Rounding

Practical rule: express the uncertainty with one or two significant figures, and round the measured value to the same decimal place as the uncertainty. Avoid:

g = (9.813742 ± 0.057381) m/s²

Prefer:

g = (9.81 ± 0.06) m/s²

And always state what the value after the ± sign represents: in this guide, unless stated otherwise, it is a standard uncertainty (one standard deviation of doubt), not a guaranteed interval and not a 95% confidence interval (see Part 16).

14.4 Graphs

A results graph includes: a title; labeled axes; units; visible experimental points (not just a smoothed curve); error bars when relevant; and a clear distinction between the data, the theoretical model and the computed fit (FizziQ Web, for instance, displays the fit equation next to the experimental points - the two must not be conflated in the report).

Part 15 - Comparing a measurement with an accepted value

15.1 The intuitive comparison

In introductory courses, the comparison stays qualitative: you weigh the observed gap between the measurement and the accepted value against the spread of the series, the estimated uncertainty, and the conditions of the experiment. If the gap is of the same order as the uncertainty, nothing lets you claim a disagreement; if it is much larger, an explanation must be sought.

US classrooms often quantify this with percent error. Percent error is a fine descriptive number, but it cannot validate a measurement by itself, because it ignores the uncertainty: a 2% error can be excellent (if your uncertainty is 5%) or alarming (if your uncertainty is 0.1%).

15.2 Comparing in units of the uncertainty

The better tool - standard in college labs and within reach of AP/IB students - is to count how many standard uncertainties separate the measurement from the accepted value:

z = |x_meas − x_ref| / u(x)

where x_meas is the measured value, x_ref the accepted (reference) value and u(x) the standard uncertainty of the measurement result. Common convention of interpretation: if z < 2, the result is considered consistent with the accepted value; if z ≥ 2, the result deserves closer scrutiny (and a clear disagreement sets in beyond about 3). When the reference value itself carries a non-negligible uncertainty u(x_ref), replace u(x) in the denominator by √(u(x)² + u(x_ref)²).

Example: a free-fall measurement gives g = (9.74 ± 0.05) m/s², to be compared with the local accepted value 9.81 m/s². z = |9.74 − 9.81|/0.05 = 1.4 < 2: the result is consistent with the accepted value.

15.3 An inconsistency is not necessarily a failure

A large z is information, not a penalty. It may reveal: an underestimated uncertainty (forgotten sources); a bias (calibration, zero, alignment); an insufficient procedure; a saturated sensor; an outlier that contaminated the mean; an incomplete physical model (friction wrongly neglected); or a genuine experimental difference. Scientific inquiry begins precisely there: finding out which of these explanations holds.

Part 16 - Measurement uncertainty and confidence intervals

16.1 Standard uncertainty

The standard uncertainty u(x) characterizes the doubt attached to a measured value, expressed as one standard deviation. The notation:

x ± u(x)

means “value x, with a doubt characterized by u(x)”. It must not be automatically presented as a 95% confidence interval: the interval [x − u, x + u] does not, in general, carry a “95% probability” - nor any simple probability at all without further assumptions.

16.2 Confidence interval

The confidence interval belongs to statistical inference: it serves, for example, to estimate the mean of a measurement process from a sample of n observations. (Students who have taken AP Statistics will recognize the construction.)

Going further - first-year college. Under suitable assumptions (independent observations from a roughly Gaussian process), a two-sided 95% confidence interval for the process mean is written:

x̄ ± t₍ₙ₋₁; 0.975₎ × s/√n

where x̄ is the sample mean, s its sample standard deviation, n the number of observations, and t₍ₙ₋₁; 0.975₎ the Student’s t coefficient, which depends on n (it is about 2.26 for n = 10 and approaches 1.96 as n grows large). This formula is not expected in high school.

16.3 The correct interpretation

The 95% confidence level characterizes the procedure used to build the interval, not any particular interval: if the sampling and construction were repeated a great many times, about 95% of the intervals so built would contain the parameter sought. Avoid the simplistic statement “there is a 95% chance that the true value lies in this interval”: once the interval is computed, the parameter’s value either lies in it or it does not - it is the method that succeeds 95 times out of 100, not the interval that “probably contains”.

Box - Uncertainty vs. confidence interval. The standard uncertainty u(x) is one standard deviation of doubt attached to a measurement result; x ± u(x) is not a 95% interval. The confidence interval is a statistical-inference tool built from a sample, with a chosen confidence level, which characterizes the reliability of the estimation procedure. The two notions meet in simple cases, but they are not interchangeable.

16.4 Required distinctions

The confidence interval must not be confused with any of the following: the sensor’s measuring range (Part 8); a tolerance interval (a manufacturing specification); the resolution (Part 6); the interval x ± u(x) (standard uncertainty); a guaranteed maximum error; an expanded uncertainty (a standard uncertainty multiplied by a coverage factor, mentioned here without making it a central tool); or the range of the observations (Part 3).

16.5 Simulation activity

For late high school (AP Statistics crossover) or first-year college: (1) collect - or simulate in Python, for instance in FizziQ Web’s built-in interpreter - many small samples of measurements of the same process; (2) compute each sample’s mean; (3) build a confidence interval for each; (4) observe that the intervals differ from sample to sample; (5) examine the proportion of intervals that contain the reference value, and compare it with the chosen confidence level. This activity makes the frequentist interpretation of 16.3 concrete.

Part 17 - The sensors as case studies

Rather than a catalog, four families of experiments, each illustrating a cluster of concepts.

17.1 The accelerometer

The ideal sensor to start with: always available, fast, and endowed with a natural reference (g). It illustrates: variability and noise (sensor at rest); effective resolution (fluttering decimals); bias and calibration (+g / −g / horizontal orientations, Part 10.5); sampling rate (impacts and vibrations, Part 7); saturation (hard shaking); and comparison across devices (Part 9). For what this sensor physically measures, see the section on the accelerometer in the guide Smartphone Sensors: How They Work, Accuracy, and Scientific Uses.

17.2 The microphone and the sound level meter

They illustrate: the difference between absolute measurement (uncertain without calibration) and relative measurement (robust); calibration and its domain of validity (Part 10.6); the influence of frequency (the microphone’s response is not uniform); orientation and position; software processing (automatic gain, noise reduction); saturation; and usage limits (a smartphone measurement is never a regulatory measurement). The complete guide to sound analysis develops all these points.

17.3 The magnetometer

It illustrates spectacularly: the influence of the environment (steel structures, magnets, electronics); reproducibility (the same measurement changes from one room to another); bias and its correction (the figure-eight calibration offered in FizziQ); the effect of orientation; and comparison between smartphones. Full three-dimensional correction models are professional-level material and beyond the scope of this guide.

17.4 Video analysis

FizziQ’s video motion analysis is a metrology digest: spatial calibration (the pixel-to-meter conversion via a reference object, whose uncertainty propagates to every position); parallax and perspective (film perpendicular to the plane of motion); clicking uncertainty (a few pixels per click); frame rate and time resolution (Part 12.4); the uncertainty of a position, then of a velocity computed by differences; and the amplification of noise when computing acceleration (differentiating twice strongly amplifies clicking fluctuations - which is why FizziQ offers a quadratic fit that smooths this computation).

17.5 The other sensors at a glance

Sensor	Quantity obtained	Main limitation	Simple experiment	Relative or absolute	Suitable use	Main precaution
GPS	Position, altitude, speed	Variable position accuracy (often several meters), slow rate	Trace a route and compare with the map	Mostly relative (changes)	Distances and speeds over long trips	Outdoors, clear sky, wait for stabilization
Barometer	Atmospheric pressure	Missing on some devices; drifts with weather	Pressure change between floors	Relative (differences)	Elevation changes, trends	Compare changes, not isolated values
Gyroscope	Rotation rate	Zero bias; drift of the integrated angle; limited range	Spinning on an office chair	Absolute short-term	Rotational motion	Avoid very fast spins (saturation)
Light meter	Illuminance	Response depends on model and angle	1/d² law with a lamp	Mostly relative	Illuminance comparisons	Keep the sensor’s orientation constant
Camera (video)	Time-stamped positions	Frame rate, perspective	Filmed free fall	Absolute after spatial calibration	Kinematics	Calibrate the scale, film head-on
Colorimeter	Color components, absorbance	Sensitive to ambient lighting	Tracking a decolorization	Relative	Comparisons, kinetics	Constant lighting, neutral background

Part 18 - Twelve classroom activities with FizziQ

Activity 1 - Are all the decimals significant?

Question: what are the displayed decimals worth? - Level: intro high school (9th-10th). - Duration: 30 min. - Equipment: one smartphone. - FizziQ tool: absolute acceleration (or magnetometer), 30 s recording at rest. - Procedure: set the device down, record, zoom into the graph, read the amplitude of the fluctuations. - Expected data: a noisy series around a stable value. - Analysis: range, choice of rounding. - Concept learned: display resolution ≠ effective resolution. - Main uncertainty sources: sensor noise, vibrations. - Expected result: only one or two stable decimal places. - Limits: strongly model-dependent. - Extension: compare several device models.

Activity 2 - Building a distribution of measurements

Question: how do repeated measurements distribute themselves? - Level: intro high school. - Duration: 1 h. - Equipment: the class’s smartphones. - Tool: instant capture (the measurement table builds itself automatically); pooling via QR codes; histogram via the Statistics mode, the data table or Python. - Procedure: each group measures the same quantity 10 times; the class’s data are pooled. - Analysis: histogram, mean, standard deviation. - Concept: variability, describing a series. - Expected result: an approximate bell-shaped distribution. - Limits: device-to-device effects mixed into the spread. - Extension: separate the histograms by device (→ Activity 6).

Activity 3 - Spread or uncertainty of the mean?

Question: what do you gain by measuring more? - Level: honors / AP / IB (11th-12th). - Duration: 1 h. - Equipment: one smartphone. - Tool: instant captures, data table (mean and standard-deviation functions). - Procedure: series of 5, 10 and 30 measurements of the same quantity. - Analysis: compute x̄, s and s/√n for each series. - Concept: the s vs. s/√n distinction. - Expected result: s stable, s/√n decreasing. - Uncertainty sources: possible drift during the long series. - Limits: correlated values if the signal is filtered. - Extension: plot u_A against n and compare with the 1/√n law.

Activity 4 - Precise but wrong?

Question: is a tightly clustered result necessarily accurate? - Level: intro high school / honors. - Duration: 45 min. - Equipment: two smartphones with contrasting behavior (or a deliberately biased series: phone slightly tilted). - Tool: accelerometer, data table. - Procedure: measure g with both configurations, compare spread and distance to the accepted value. - Concept: accuracy vs. precision, bias. - Expected result: one precise-but-offset series, one scattered-but-accurate-on-average series. - Limits: requires genuinely contrasting devices. - Extension: correct the bias (→ Activity 5).

Activity 5 - Calibrating an accelerometer

Question: can gravity be used to characterize and correct a sensor’s bias? - Level: honors / AP (11th-12th). - Duration: 1 h. - Equipment: one smartphone, a well-leveled surface and a vertical support. - Tool: accelerometer (axis by axis), data table. - Procedure: record the studied axis in the +g, −g and horizontal positions; compute the half-sum (zero) and half-difference (scale). - Analysis: estimate the zero offset and the scale factor; apply the correction in the data table. - Concept: calibration, zero, scale factor, domain of validity. - Expected result: small but measurable offsets. - Uncertainty sources: actual levelness and verticality of the positions. - Extension: check the correction on a new measurement.

Activity 6 - Comparing several smartphones

Question: why don’t all devices give the same thing? - Level: all high school levels. - Duration: 1 h. - Equipment: several smartphones. - Tool: the same instrument on all, a written shared procedure, pooling via QR codes. - Procedure: each device repeats the same measurement; compare within-device spreads and between-device means. - Concept: repeatability vs. reproducibility. - Expected result: between-device spread larger than within-device spread. - Limits: possible confusion with operator effects. - Extension: swap the operators to isolate their effect.

Activity 7 - Detecting saturation

Question: how do you recognize that a sensor is overwhelmed? - Level: intro high school. - Duration: 30 min. - Equipment: one smartphone, a source of increasing intensity (shaking, rising sound). - Tool: accelerometer or sound level meter, continuous recording. - Procedure: intensify the phenomenon gradually, watch for the plateau. - Concept: measuring range, saturation, “stable but wrong”. - Expected result: a plateauing curve, flattened peaks. - Precaution: reasonable sound volume, protect the device. - Extension: compare the ceilings of several devices.

Activity 8 - The effect of the sampling rate

Question: what is lost when you sample too slowly? - Level: AP / IB (11th-12th). - Duration: 45 min. - Equipment: one smartphone. - Tool: accelerometer, sampling-rate setting in FizziQ’s Settings. - Procedure: record a slow motion, a vibration and an impact, at two different acquisition rates. - Concept: sampling rate, fast phenomena. - Expected result: the impact is truncated or missed at the low rate. - Limits: maximum rate depends on the model. - Extension: the Shannon-Nyquist box (12th grade / college).

Activity 9 - Measuring a speed and its uncertainty

Question: how do you report a speed with its uncertainty? - Level: AP / IB (12th). - Duration: 1 h 30. - Equipment: a moving object (a person walking, a ball), a tape measure, a smartphone. - Tool: stopwatch (or acoustic stopwatch) and data table; or video analysis. - Procedure: measure d and t several times; evaluate u(d) (Type B, tape measure) and u(t) (Type A over the repetitions + Type B from the resolution). - Analysis: v = d/t, combination of the relative uncertainties (supplied rule, Part 13.3). - Concept: simple combined uncertainty, dominant source. - Expected result: the duration usually dominates the uncertainty. - Extension: improve the weak link and re-measure.

Activity 10 - Comparing with an accepted value

Question: is my measurement of g consistent with the accepted value? - Level: AP / IB (12th). - Duration: 1 h. - Equipment: depends on the method (filmed free fall, pendulum, accelerometer at rest). - Tool: video analysis or accelerometer, data table. - Procedure: measure g, evaluate u(g), compute z = |g_meas − g_ref|/u(g). - Concept: comparison in units of the uncertainty, consistency. - Expected result: z < 2 with a careful procedure; otherwise, discussion of causes (15.3). - Limits: the conclusion depends on the quality of the evaluation of u(g). - Extension: compare the z of several methods for measuring g.

Activity 11 - Calibrating a sound level meter

Question: can a smartphone’s sound level meter be made accurate? - Level: honors / AP. - Duration: 1 h. - Equipment: a reference sound level meter (or one device taken as a common reference), a stable sound source. - Tool: the FizziQ sound level meter and its calibration setting (offset in dB). - Procedure: strictly identical conditions (position, orientation, source, frequency, duration); record the discrepancy; apply the correction; check at another level and another frequency. - Concept: calibration, domain of validity of a correction. - Expected result: correction effective near the calibration point, less so elsewhere. - Limits: without a reference instrument, only a relative inter-comparison is possible.

Activity 12 - Building confidence intervals

Question: what does “95% confidence” mean? - Level: AP Statistics crossover / first-year college. - Duration: 1 h 30. - Equipment: a smartphone or a computer. - Tool: real exported data (CSV/Python) or a simulation in FizziQ Web’s Python interpreter. - Procedure: build many small samples, compute each mean and its Student’s t interval. - Analysis: proportion of intervals containing the reference. - Concept: the frequentist interpretation of the confidence interval. - Expected result: a proportion close to 95%. - Limits: assumptions (independence, approximate normality). - Extension: vary n and the confidence level.

Part 19 - When is a smartphone good enough?

19.1 Suitable uses

The smartphone is often sufficient to: observe a phenomenon; measure a change; compare two situations; determine an order of magnitude; test a physical relationship (proportionality, inverse-square law…); run a school experiment; study variability itself; compare procedures; work in the field; and learn to process real data.

19.2 Uses requiring a specialized instrument

A professional instrument is required when the goal is: regulatory; medical; legal; contractual; safety-related; or when it demands metrological traceability; when the quantity falls outside the sensor’s range; when it is smaller than the smartphone’s noise; when the phenomenon is too fast for the acquisition; or when results must be comparable between laboratories under strict requirements.

19.3 Decision grid

Question to ask	Smartphone suitable	Specialized instrument preferable
Is the goal to compare two situations or track a change?	Yes, very well suited	-
Is an exact absolute value needed?	Only after calibration, with caution	Yes if absolute accuracy is the point
Is the quantity within the sensor’s range and above its noise?	Yes	No → dedicated instrument
Is the phenomenon slower than the acquisition rate?	Yes	No → dedicated fast acquisition
Does the result carry regulatory, medical or legal weight?	No	Yes, mandatorily
Is the goal to learn the measurement process?	Ideal	-

The final criterion never changes: is the quality of the measurement sufficient to answer the scientific question being asked? An instrument is suitable not because it is digital or professional, but because its quality meets the need.

Part 20 - Common mistakes and confusions

“Six displayed decimals mean six decimals of precision.” False: the display format determines neither the effective resolution nor the uncertainty (Parts 6 and 14).

“Tightly clustered measurements must be accurate.” False: they can be precise but biased; precision says nothing about accuracy (Part 4).

“Taking more measurements removes all the errors.” False: repetition reduces the statistical uncertainty (s/√n), but does nothing to a bias shared by all the measurements (Parts 5 and 11).

“Calibrating just means zeroing.” False: zeroing only fixes an offset; calibration establishes the relationship between the readings and reference values, possibly at several points (Part 10).

“The standard deviation is the error of the measurement.” False: the standard deviation describes the spread of the observations, not the distance to the true value (Part 3).

“The uncertainty is the difference from the true value.” False: that difference is the error, which is generally unknown; the uncertainty is the estimate of the doubt attached to the result (Part 5).

“x ± u(x) is a 95% confidence interval.” False: unless stated otherwise, u(x) is a standard uncertainty, not the half-width of a 95% interval (Part 16).

“A small percent error proves the measurement is good.” Incomplete: percent error ignores the uncertainty; only the comparison in units of the uncertainty can support a claim of agreement (Part 15).

“A stable value is necessarily reliable.” False: a saturated or heavily filtered sensor can be perfectly stable and completely wrong (Part 8).

“All smartphones give the same results.” False: sensors, software processing and calibrations differ from one model and operating system to another (Part 9).

“Digital data is objective by nature.” False: it depends on the sensor, the software, the procedure and the processing - the whole chain must be understood to interpret the number (Part 1).

“A smartphone can’t be a scientific instrument.” False: it can be perfectly suited to a great many questions, provided its limits are known - which is precisely the point of this guide (Part 19).

Part 21 - Frequently asked questions

Are smartphone sensors reliable? They are reliable enough for a great many scientific experiments, especially for comparing situations and tracking changes. Their absolute accuracy, however, varies by model and is not guaranteed without calibration. The right question is not “reliable or not” but “sufficient or not for the question asked” (Part 19).

Why does a measurement vary? Because many small causes add up: electronic noise, vibrations, temperature, environment, the operator’s gesture, software rounding. This variability is universal; you study it by repeating the measurement (Part 2).

What do the digits after the decimal point mean? They reflect the display format chosen by the software, not the quality of the measurement. Only the digits that stay stable at the level of the uncertainty are significant; decimals fluttering under the noise carry no information (Parts 6 and 14).

What is the difference between precision and resolution? Resolution is the smallest step the instrument can represent or display; precision describes how tightly repeated results cluster. A high-resolution instrument can be imprecise, and vice versa (Part 6).

What is the difference between accuracy and precision? Precision describes the clustering of repeated measurements (small spread); accuracy describes the absence of bias with respect to an accepted value. A series can be precise and wrong, or scattered and accurate on average (Part 4).

What is the difference between error and uncertainty? The error is the actual - generally unknown - gap between the result and the value sought. The uncertainty is the calculable estimate of the doubt attached to the result (Part 5).

What is the standard deviation for? To quantify the spread of a series of measurements around its mean. It measures neither the error nor the accuracy: two series with the same standard deviation can be one centered on the accepted value, the other far off (Part 3).

Why repeat a measurement? To make the variability visible, detect outliers, and reduce the statistical uncertainty of the mean, which shrinks like s/√n (the standard error). Repetition, however, corrects no bias (Parts 3 and 11).

How many measurements should you take? There is no magic number: about ten measurements already give a good picture of the spread, and the uncertainty of the mean then decreases only slowly (as 1/√n). Ten measurements under a stable procedure beat a hundred under changing conditions (Part 11).

What is a Type A uncertainty? A standard uncertainty evaluated statistically from a series of repeated measurements: u_A = s/√n under the right conditions - the formal name for what lab manuals call the random or statistical uncertainty (Part 11).

What is a Type B uncertainty? A standard uncertainty evaluated from non-statistical information: resolution, tolerance, specification, reference. For an assumed uniform interval of half-width a, use u_B = a/√3 - the general framework behind the ”± half the smallest division” rule (Part 12).

How do you combine several uncertainties? For independent contributions expressed as standard uncertainties, add them in quadrature (root-sum-square): u_c = √(u₁² + u₂² + …). For a computed quantity such as v = d/t, it is the relative uncertainties that combine this way (Part 13).

How do you calibrate a smartphone? By comparing its readings with a reference under controlled conditions, then correcting: through the app’s setting when one exists (sound-level-meter and magnetometer calibration in FizziQ) or by correcting the data in the table. The correction is only valid within its calibration domain (Part 10).

Can you calibrate an accelerometer with gravity? Yes: by pointing the sensor’s axis successively up, down and horizontally, you estimate the zero offset and the scale factor from the readings near +g, −g and 0 (Part 10.5, Activity 5).

Why do two smartphones give different results? Because they carry different sensors, different factory calibrations and different software processing. These discrepancies are normal, and they make an excellent vehicle for teaching reproducibility (Part 9).

How do you know a sensor is saturated? By the characteristic signs: a value that plateaus, an identical repeated maximum, flattened peaks, a display that stops rising while the phenomenon intensifies. A saturated value can be very stable and still wrong (Part 8).

What is the sampling rate? The number of values acquired per second. Too low, it misses peaks, distorts fast phenomena and can create false slow signals (Part 7).

How do you choose significant figures? Express the uncertainty with one or two significant figures, then round the value to the same decimal place: g = (9.81 ± 0.06) m/s², not g = (9.813742 ± 0.057381) m/s² (Part 14).

What is a confidence interval? An interval built from a sample by a statistical procedure whose confidence level (say 95%) describes the procedure’s success rate: over a great many repetitions, about 95% of the intervals so built would contain the parameter sought (Part 16).

What is the difference between a confidence interval and measurement uncertainty? The standard uncertainty is one standard deviation of doubt attached to a result; the confidence interval is a statistical-inference tool with a chosen confidence level. The notation x ± u(x) is not a 95% interval (Part 16).

How do you compare a measurement with an accepted value? Qualitatively in introductory courses (gap weighed against spread), then quantitatively by counting how many standard uncertainties separate measurement and reference: below 2, the result is considered consistent with the accepted value. Percent error alone cannot settle the question, because it ignores the uncertainty (Part 15).

When should you use a professional instrument? As soon as the stakes are regulatory, medical, legal, contractual or safety-related, or when the measurement demands traceability, a range, a fineness or a speed beyond the smartphone’s reach (Part 19).

Can FizziQ be used to teach uncertainty? Yes: the app gives access to the sensors, records series, automatically builds tables of repeated measurements, displays distributions, computes means and standard deviations in its data table, exports to CSV and Python, and documents everything in an experiment notebook - the whole chain needed for a complete measurement workflow. The concepts taught remain valid with any other instrument.

Conclusion

At the end of this journey, a few ideas deserve to stick. A measurement is never just a number: it is a value, a unit, a procedure, conditions - and an evaluated doubt. Displayed decimals guarantee nothing: the quality of a result is judged by its variability, its biases and its uncertainty, not by its display format. Repeating a measurement lets you study variability, and the mean is not enough without information about the spread; a precise result can be biased, and calibration is what characterizes or corrects those biases. Uncertainty expresses the doubt attached to the result - it is neither the error, nor the standard deviation, nor a confidence interval, which is a distinct statistical tool with its own interpretation.

The smartphone is an excellent tool for learning all of this, precisely because it quickly produces abundant, imperfect data: visible noise, real biases, reproducible saturations, differences between devices. FizziQ turns that data into a complete scientific workflow - measure, repeat, display, compute, compare, document. And the final criterion never changes: an instrument is suitable not because it is digital or professional, but because its quality is sufficient to answer the question being asked.

A reliable measurement is not the one that displays the most digits. It is the one whose origin, variability, limits and uncertainty you understand.

To go further on fizziq.org: the guide Smartphone Sensors: How They Work, Accuracy, and Scientific Uses, the complete guide to sound analysis, the article Speed of sound - seven activities with a smartphone, the activity catalog, and the FizziQ and FizziQ Web user guides at fizziq.org/documentation.

Sources and references

International Vocabulary of Metrology (VIM, JCGM 200) for the definitions of measurand, trueness, precision and calibration; Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100), whose framework NIST adopts in the United States (NIST Technical Note 1297), for Type A and Type B evaluations and the quadrature combination; the AP Physics and AP Chemistry laboratory guidance and the IB Diploma Programme sciences guides for the treatment of uncertainty, error bars and significant figures expected in high school; university introductory-laboratory manuals for the standard deviation vs. standard error distinction and the comparison of results in units of the uncertainty; FizziQ and FizziQ Web documentation (fizziq.org/documentation) for the operation of the instruments cited.

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