We consider a random variable x and a data set S = {x₁, x₂, …, x_n} of size n which contains possible values of x. The data set can represent either the population being studied or a sample drawn from the population. The mean is the statistic used most often to characterize the center of the data in S. We now consider the following commonly used measures of variability of the data around the mean, namely the standard deviation, variance, squared deviation and average absolute deviation.

In addition we also explore three other measures of variability that are not linked to the mean, namely the median absolute deviation, range and inter-quartile range.

Of these statistics the variance and standard deviation are most commonly employed.

Excel Functions: If R is an Excel range which contains the data elements in S then the Excel function which calculates each of these statistics is shown in Figure 1. Functions marked with an asterisk are supplemental functions found in the Real Statistics Resource Pack, although equivalent formulas in standard Excel are described later.

Figure 1 – Measures of Variability

Observation: These functions ignore any empty or non-numeric cells.

Variance

Definition 1: The variance is a measure of the dispersion of the data around the mean. Where S represents a population the population variance (symbol σ²) is calculated from the population mean µ as follows:

Where S represents a sample the sample variance (symbol s²) is calculated from the sample mean x̄ as follows:

The reason the expression for the population variance involves division by n while that of the sample variance involves division by n – 1 is explained in Property 3 of Estimators, where division by n – 1 is required to obtained an unbiased estimator of the population variance.

Excel Function: The sample variance is calculated in Excel using the worksheet function VAR. The population variance is calculated in Excel using the function VARP. In Excel 2010/2013 the alternative forms of these functions are VAR.S and VAR.P.

Example 1: If S = {2, 5, -1, 3, 4, 5, 0, 2} represents a population, then the variance = 4.25.

This is calculated as follows. First, the mean = (2+5-1+3+4+5+0+2)/8 = 2.5, and so the squared deviation SS = (2–2.5)² + (5–2.5)² + (-1–2.5)² + (3–2.5)² + (4–2.5)² + (5–2.5)² + (0–2.5)² + (2–2.5)² = 34. Thus the variance = SS/n = 34/8 = 4.25

If instead S represents a sample, then the mean is still 2.5, but the variance = SS/(n–1) = 34/7 = 4.86.

These can be calculated in Excel by the formulas VARP(B3;B10) and VAR(B3:B10), as shown in Figure 2.

Figure 2 – Examples of measures of variability

Observation: When data is expressed in the form of frequency tables then the following properties are useful.

Property 1: If x̄ is the mean of the sample S = {x₁, x₂, …, x_n}, then the sample variance can be expressed by

Property 2: If µ is the mean of the population S = {x₁, x₂, …, x_n}, then the population variance can be expressed by

Standard Deviation

Definition 2: The standard deviation is the square root of the variance. Thus the population and sample standard deviations are calculated respectively as follows:

Excel Function: The sample standard deviation is calculated in Excel using the worksheet function STDEV. The population standard deviation is calculated in Excel using the function STDEVP. In Excel 2010/2013 the alternative forms of these functions are STDEV.S and STDEV.P.

Example 2: If S = {2, 5, -1, 3, 4, 5, 0, 2} is a population, then the standard deviation = square root of the population variance =  = 2.06

If S is a sample, then the sample standard deviation = square root of the sample variance =  = 2.20

These are the results of the formulas STDEVP(B3:B10) and STDEV(B3:B10), as shown in Figure 2.

Real Statistics Functions: The Real Statistics Resource Pack furnishes the following array functions:

VARCOL(R1) = a row range which contains the sample standard variances of each of the columns in R1

STDEVCOL(R1) = a row range which contains the sample standard deviations of each of the columns in R1

VARROW(R1) = a column range which contains the sample standard variances of each of the rows in R1

STDEVROW(R1) = a column range which contains the sample standard deviations of each of the rows in R1

Example 3: Use the VARCOL and STDEVCOL functions to calculate the sample variance and standard deviation of each of the columns in the range L4:N11 of Figure 3.

The formula =VARCOL(J4:L11) produces the first result (in range J15:L15), while the formula =STDEVCOL(J4:L11) produces the second result (in range J16:L16). Remember that after entering either of these formulas you must press Ctrl-Shft-Enter.

Figure 3 – Sample Variance and Standard Deviation by Column

Property 3: If the population {x₁, x₂, …, x_n} has mean µ_x and standard deviation σ_x and the population {y₁, y₂, …, y_m} has mean µ_y and standard deviation σ_y, then the variance of the combined population is

Thus if µ_x = µ_y the combined population variance would be

Property 4: If the sample {x₁, x₂, …, x_n} has mean x̄ and standard deviation s_x and the sample {y₁, y₂, …, y_m} has mean ȳ and standard deviation s_y, then the variance of the combined sample is

Thus if x̄ = ȳ, the combined sample variance would be

Example 4: Find the mean and variance of the sample which results from combining the two samples {3, 4, 6, 7} and {6, 1, 5}.

Figure 4 – Calculation of combined mean and standard deviation

The data in the two samples is given in the range B3:C7 of Figure 4. From these, the mean, variance and standard deviation are calculated for each of the two samples (ranges B12:B15 and C12:C15). Using Property 4, we can calculate the mean and variance of the combined sample (D13 and D14).

If we simply combine the two samples we obtain the data in the range F3:F10, from which we can calculate the mean, variance and standard deviation in the normal way (range D12:D18). As we can see the results are the same.

Observation: In practice instead of using Property 3 and 4, we use the approach shown in the following example, especially since it can be applied to more than two samples or populations.

Example 5: Find the mean and variance of the sample which results from combining the three samples shown in range A3:D6 of Figure 5.

Figure 5 – Calculation of combined mean and variance

We have three samples whose total sample size is 58 (cell B7), calculated via =SUM(B4:B6). The sum of the elements in each sample can be calculated from the mean as shown in range F4:F6. E.g. the sum of all the data elements in sample 1 is 276 (cell F4), calculated via the formula =B4*C4. Thus the sum of all the elements in all three sample is 786 (cell F7), calculated via the formula =SUM(F4:F6). The mean of the combined sample is therefore 13.5517 (cell C7), calculated via the formula =F7/B7.

The calculation of the combined variance is similar. The key is to first find the sum of the squares of all the elements in each sample. These are given in range I4:I6. E.g. the sum of the squares of all the elements in sample 1 is 5512 (cell I4), calculated by =G4+H4 (using Property 1), where G4 contains the formula =B4*C4^2 and H4 contains =(B4-1)*D4. Thus the sum of squares of all the elements in the combined sample is 19,832 (cell I7), calculated by =SUM(I4:I6). Finally, the variance for the combined sample is 161.059 (cell D7), calculated by =(I7-B7*C7^2)/(B7-1), based on Property 1. The standard deviation is therefore 12.6909.

Squared Deviation

Definition 3: The squared deviation (symbol SS for sum of squares) is most often used in ANOVA and related tests. It is calculated as

Excel Function: The squared deviation is calculated in Excel using the worksheet function DEVSQ.

Example 6: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the squared deviation = 34. This is the same as the result of the formula DEVSQ(B3:B10) as shown in Figure 2.

Average Absolute Deviation

Definition 4: The average absolute deviation (AAD) of data set S is calculated as

Excel Function: The average absolute deviation is calculated in Excel using the worksheet function AVEDEV.

Example 7: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the average absolute deviation = 1.75. This is the same as the result of the formula AVEDEV(B3:B10) as shown in Figure 2.

Median Absolute Deviation

Definition 5: The median absolute deviation (MAD) of data set S is calculated as

Median {|x_i – | : x_i in S}

where  = median of the data elements in S.

Excel Formula: If R is a range which contains the data elements in S then the MAD of S can be calculated in Excel by the array formula:

=MEDIAN(ABS(R-MEDIAN(R)))

Even though the value is presented in a single cell it is essential that you press Ctrl-Shft-Enter to obtain the array value, otherwise the result won’t come out correctly. This function only works properly when R doesn’t contain any empty cell or cell with a non-numeric value.

Alternatively, you can use the supplemental function MAD(R) which is contained in the Real Statistics Resource Pack. This function works properly even when R contains empty cells and/or cells with non-numeric values. You don’t need to press Ctrl-Shft-Enter  to use this function.

Example 8: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the median absolute deviation = 2 since S = {-1, 0, 2, 2, 3, 4, 5, 5}, and so the median of S = (2+3)/2 = 2.5. Thus MAD = the median of {3.5, 2.5, 0.5, 0.5, 0.5, 1.5, 2.5, 2.5} = {0.5, 0.5, 0.5, 1.5, 2.5, 2.5, 2.5, 3.5}, i.e. (1.5+2.5)/2 = 2.

You can achieve the same result using the supplemental formula =MAD(E3:E10) as shown in Figure 2.

Observation: This metric is less affected by extremes in the tails because the data in the tails have less influence on the calculation of the median than they do on the mean.

Range

Definition 6: The range of a data set S is a crude measure of variability and consists simply of the difference between the largest and smallest values in S.

Excel Formula: If R is a range which contains the data elements in S then the range of S can be calculated in Excel by the formula:

=MAX(R) – MIN(R)

Alternatively, you can use the supplemental function RNG(R) which is contained in the Real Statistics Resource Pack.

Example 9: If S = {2, 5, -1, 3, 4, 5, 0, 2}, the range = 5 – (-1) = 6. You can achieve the same result using the supplemental formula =RNG(E3:E10) as shown in Figure 2.

Inter-quartile Range

Definition 7: The inter-quartile range (IQR) of a data set S is calculated as the 75% percentile of S minus the 25% percentile. The IQR provides a rough approximation of the variability near the center of the data in S.

Excel Formula: If R is a range which contains the data elements in S then the IQR of S can be calculated in Excel by the formula:

=QUARTILE(R, 3) – QUARTILE(R, 1)

In Excel 2010/2013 there is a new version of the quartile function called QUARTILE.EXC. An alternative version of IQR is therefore

=QUARTILE.EXC(R, 3) – QUARTILE.EXC(R, 1)

See Ranking Functions in Excel for further information about the QUARTILE and QUARTILE.EXC functions. Alternatively, you can calculate the inter-quartile range via the supplemental function IQR(R, b) which is contained in the Real Statistics Resource Pack. When b = FALSE (default), the first version of IQR is returned, while when b = TRUE the second version is returned.

Example 10: If S = {2, 5, -1, 3, 4, 5, 0, 2}, then the first version of IQR = 4.25 – 1.5 = 2.75, while the second version is IQR = 4.75 – 0.5 = 4.25. You can achieve the same result using the Real Statistics formulas =IQR(B3:B10) and =IQR(B3:B10,TRUE), as shown in Figure 2.

Observation: The variance, standard deviation, average absolute deviation and median absolute deviation measure both the variability near the center and the variability in the tails of the distribution which represents the data. The average absolute deviation and median absolute deviation do not give undue weight to the tails. On the other hand, the range only uses the two most extreme points and the interquartile range only uses the middle portion of the data.

Coefficient of Variation

Definition 8: The coefficient of variation (aka the coefficient of variability), V (or CV), of the data set S is calculated as

V = s/x̄

Since s and x̄ have the same units of measurement, V has no units of measurement. This statistic only makes sense for ratio scale data. The higher the value of V the more dispersion there is.

Clearly, the coefficient of variation is only defined when the mean is not zero.

Excel Formula: If R is a range which contains the data elements in S then the coefficient of variation for S can be calculated in Excel by the formula:

=STDEV.S(R)/AVERAGE(R)

The population version of V is σ/μ which can be calculated in Excel by the formula

=STDEV.P(R)/AVERAGE(R)

Example 11: If  S = {2, 5, -1, 3, 4, 5, 0, 2} represents a sample, then, as we can see from Example 1, the coefficient of variation is

V = s/x̄ = 2.203892/2.5 = 88.16%

Example 12: Stock A has an expected return of 12% with a standard deviation of 9% and stock B has an expected return of 8% with a standard deviation of 5%. Use the coefficient of variation to determine which is the better investment.

Since V_A = .09/.12 = .75 and V_B = .05/.08 = .625, stock B is considered to be the better investment since its relative risk (equal to its coefficient of variation) is lower.