Standard Deviation

Standard deviation is a statistical quantity that measures the dispersion of a set of values around their mean, indicating how much measurements are grouped or spread out.

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How to measure it in class

With the FizziQ app, it is possible to calculate the standard deviation of a series of atmospheric pressure measurements.

Steps:

Open FizziQ and select the pressure sensor. Record pressure for 60 seconds with the smartphone placed on the table, without touching it.
Export the data to a spreadsheet. Calculate the mean and standard deviation of the series of pressure values obtained.
Repeat the experiment by placing the smartphone in a more disturbed location (near a door, on a window ledge) and recalculate the standard deviation.
Compare the two standard deviations obtained. The first represents the intrinsic noise of the sensor, the second includes environmental disturbances. Discuss the meaning of this difference.

Scientific activities on this topic

Several experiments easily achievable with a smartphone, tablet, or computer allow calculating and interpreting the standard deviation of experimental measurements.

1: Uncertainty - Analyzing measurement uncertainties: https://www.fizziq.org/en/activities/uncertainty/
2: Evaluating sensor quality - Analysis of sensor precision: https://www.fizziq.org/en/activities/evaluate-the-quality-of-a-sensor/
3: Pendulum period - Build a device to precisely determine the period of a pendulum: https://www.fizziq.org/en/activities/period-of-a-pendulum/

Learn more

History of standard deviation

The concept of standard deviation was formalized by Karl Pearson in 1894, although early studies on measurement dispersion date back to Carl Friedrich Gauss in the early 19th century. Gauss developed the least squares method to fit astronomical data and introduced the normal distribution, sometimes called the Gaussian curve, whose central parameter is the standard deviation.

Standard deviation and normal distribution

For a Gaussian distribution, standard deviation has a precise geometric interpretation. It corresponds to the inflection point of the bell curve. Approximately 68.3% of values are found in the interval [mean - sigma; mean + sigma], 95.4% in [mean - 2sigma; mean + 2sigma], and 99.7% in [mean - 3sigma; mean + 3sigma]. This is the 68-95-99.7 rule.

Standard deviation of the mean

When calculating the mean of n independent measurements, the uncertainty on this mean is smaller than the individual standard deviation. The standard deviation of the mean equals sigma/sqrt(n). That is why repeating measurements improves the precision of the final result. Going from 4 to 16 measurements divides uncertainty by 2.

Applications in quality control

In industry, the Six Sigma method aims to reduce variability in production processes. The goal is for manufacturing tolerance to correspond to six standard deviations on either side of the target value, which guarantees a defect rate of less than 3.4 per million parts produced.

Formula

The standard deviation of a sample of n measurements is calculated by:

sigma = sqrt[ sum((xi - x_bar)^2) / (n - 1) ]

Meaning: sigma: standard deviation (in the unit of the measured quantity) xi: value of measurement i x_bar: arithmetic mean of measurements n: total number of measurements sum: sum over all measurements from i = 1 to n

Application examples

Lap times of an F1 driver: a low standard deviation means consistent driving
Class grades: a large standard deviation indicates very varied levels among students
Weights of candy packets in production: standard deviation controls uniformity of filling
Daily temperature of a city: annual standard deviation reflects climate amplitude
Dart throws of a player: a low standard deviation means grouped throws

FAQ

Q: What is the difference between standard deviation and variance? A: Variance is the square of standard deviation. Standard deviation is preferred because it is expressed in the same unit as the measured quantity, which facilitates interpretation.

Q: Why divide by (n-1) instead of n? A: Dividing by (n-1) instead of n corrects a statistical bias. With n-1, a better estimate of the actual dispersion of the population is obtained from a limited sample. This is Bessel’s correction.

Q: What does a standard deviation of zero mean? A: A zero standard deviation means all measurements are identical, which is very rare in practice. If this happens, it is often because the instrument does not have sufficient resolution to detect variations.

Q: How to interpret standard deviation in class? A: As a general rule, approximately 68% of measurements are within one standard deviation of the mean, and approximately 95% within two standard deviations, if measurements follow a normal distribution.

Q: Is standard deviation always relevant? A: Standard deviation implicitly assumes a symmetric distribution. For very asymmetric distributions or those with outliers, other indicators such as interquartile range may be more appropriate.

Variance - Mean - Uncertainty - Precision - Measurement noise - Normal distribution - Random error