2.2.5 - Measures of Spread

Variance and standard deviation are measures of variability. The standard deviation is the most commonly used measure of variability when data are quantitative and approximately normally distributed. When computing the standard deviation by hand, it is necessary to first compute the variance. The standard deviation is equal to the square root of the variance. Here, you will learn how to compute these values by hand. After this lesson, you will always be computing standard deviation using software such as Minitab Express.

Standard Deviation
Roughly the average difference between individual data values and the mean. The standard deviation of a sample is denoted as \(s\). The standard deviation of a population is denoted as \(\sigma\).
Sample Standard Deviation
\(s=\sqrt{\dfrac{\sum (x-\overline{x})^{2}}{n-1}}\)

In order to compute the standard deviation for a sample we first compute deviations. The sum of the squared deviations (SS) divided by \(n-1\), this is the variance (\(s^2\)).

The square root of the variance is the standard deviation: \(\sqrt{s^2}=s\).

Deviation
An individual score minus the mean.
Sum of Squared Deviations
Deviations squared and added together. This is also known as the sum of squares or SS.
Variance
Approximately the average of all of the squared deviations; for a sample represented as \(s^{2}\).
Sum of Squares
\(SS={\sum (x-\overline{x})^{2}}\)
Sample Variance
\(s^{2}=\dfrac{\sum (x-\overline{x})^{2}}{n-1}\)

There are a number of methods for calculating the standard deviation. If you look through different textbooks or search online, you may find different formulas and procedures. To compute the standard deviation for a sample, we will use the formulas above and the following steps:

Step 1: Compute the sample mean: \(\overline{x} = \frac{\sum x}{n}\).

Step 2: Subtract the sample mean from each individual value: \(x-\overline{x}\), these are the deviations.

Step 3: Square each deviation: \((x-\overline{x})^{2}\), these are the squared deviations.

Step 4: Add the squared deviations: \(\sum (x-\overline{x})^{2}\), this is the sum of squares.

Step 5: Divide the sum of squares by \(n-1\): \(\frac{\sum (x-\overline{x})^{2}}{n-1}\), this is the sample variance \((s^{2})\).

Step 6: Take the square root of the sample variance: \(\sqrt{\frac{\sum (x-\overline{x})^{2}}{n-1}}\), this is the sample standard deviation.

Example: Hours Spent Studying Section

Studying

A professor asks a sample of 7 students how many hours they spent studying for the final. Their responses are: 5, 7, 8, 9, 9, 11, and 13.

Step 1: Compute the mean

\(\overline{x} = \dfrac{\sum x}{n}=\dfrac{5+7+8+9+9+11+13}{7}=8.857\)

Step 2: Compute the deviations

\(x\) \(x - \overline{x}\)
5 \(5 - 8.857 = -3.857\)
7 \(7 - 8.857 = -1.857\)
8 \(8 - 8.857 = -0.857\)
9 \(9 - 8.857 = 0.143\)
9 \(9 - 8.857 = 0.143\)
11 \(11 - 8.857 = 2.143\)
13 \(13 - 8.857 = 4.143\)

Step 3: Square the deviations

\(x\) \(x - \overline{x}\) \((x-\overline{x})^{2}\)
5 \(5 - 8.857 = -3.857\) \(-3.857^{2} = 14.876\)
7 \(7 - 8.857 = -1.857\) \(-1.857^{2} = 3.448\)
8 \(8 - 8.857 = -0.857\) \(-0.857^{2} = 0.734\)
9 \(9 - 8.857 = 0.143\) \(0.143^{2} = 0.0020\)
9 \(9 - 8.857 = 0.143\) \(0.143^{2} = 0.0020\)
11 \(11 - 8.857 = 2.143\) \(2.143^{2} = 4.592\)
13 \(13 - 8.857 = 4.143\) \(4.143^{2} = 17.164\)

Step 4: Sum the squared deviations

\(SS=\sum (x-\overline{x})^{2}=14.876+3.448+0.734+.020+.020+4.592+17.164=40.854\)

The sum of squares is 40.854

Step 5: Divide by n - 1 to compute the variance

\(s^{2}=\dfrac{\sum (x-\overline{x})^{2}}{n-1}=\dfrac{40.854}{7-1}=6.809\)

The variance is 6.809

Step 6: Take the square root of the variance

\(s=\sqrt{s^{2}}=\sqrt{6.809}=2.609\)

The standard deviation is 2.609