Why is my variance so high in Excel

Why

We consider a random variablexand a data setS ={x1, x2, , xn}of size n which contains possible values ofx. The data set can represent either the population being studied or a sample drawn from the population. The meanis 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 that 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.

StatisticExcel 2007Excel 2010+SymbolPopulation VarianceVARP(R)VAR.P(R)σ2Sample VarianceVAR(R)VAR.S(R)s2Population Standard DeviationSTDEVP(R)STDEV.P(R)σSample Standard DeviationSTDEV(R)STDEV.S(R)sSquared DeviationDEVSQ(R)DEVSQ(R)SSAverage Absolute DeviationAVEDEV(R)AVEDEV(R)AADMedian Absolute DeviationMAD(R) *MAD(R) *MADRangeRNG(R) *RNG(R) *Inter-quartile RangeIQR(R, b) *IQR(R, b) *IQRCoefficient of VariationSTDEV(R)/AVERAGE(R)STDEV.S(R)/AVERAGE(R)V

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 Srepresents a population the population variance (symbol σ2) is calculated from the population meanµ as follows:

Population variance

WhereSrepresents a sample thesample variance(symbol s2) is calculated from the sample mean as follows:

Sample variance

The reason the expression for the population variance involves division by n while that of the sample variance involves division by n 1 isexplained in Property 3 ofEstimators, where division by n1is 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 deviationSS = (22.5)2+ (52.5)2+(-12.5)2+(32.5)2+(42.5)2+(52.5)2 +(02.5)2+(22.5)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/(n1) = 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.

Measures variability examples Excel

Figure 2 Examples of measures of variability

Observation: When data is expressed in the form of frequency tablesthen the following properties are useful. Click herefor the proofs of these properties.

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

image169

Property 2: If µis the mean of the population S ={x1, x2, , xn}, then the population variance can be expressed by

image173

Standard Deviation

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

Standard deviation

Excel Function:The sample standard deviation is calculated in Excel using the worksheet functionSTDEV. The population standard deviation is calculated in Excel using the functionSTDEVP.In Excel 2010/2013 the alternative forms of these functions areSTDEV.SandSTDEV.P.

Example 2: IfS={2, 5, -1, 3, 4, 5, 0, 2} is a population, then the standard deviation = square root of the population variance = \sqrt{4.25} = 2.06

IfSis a sample, thenthe sample standard deviation = square root of the sample variance = \sqrt{4.86} = 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 that contains the sample standard variances of each of the columns in R1

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

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

STDEVROW(R1) = a column range that 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.

Column variance

Figure 3 Sample Variance and Standard Deviation by Column

Property 3: If the population {x1, x2, , xn} has mean µxand standard deviation σxand the population {y1, y2, , ym} has mean µyand standard deviation σy, then the variance of the combined population is

image178

Thus if µx= µy the combined population variance would be

Combined population variance

Property 4: If the sample{x1, x2, , xn} has mean and standard deviation sxand the sample {y1, y2, , ym} has mean ȳ and standard deviation sy, then the variance of the combined sample is

Combined sample variance

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

image5015

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}.

Combined mean variance Excel

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 3and 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.

Combined mean and variance

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 samples 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 ANOVAand related tests. It is calculated as

Squared deviation

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

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

Average absolute deviation

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

Example 7: IfS={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 {|xi \tilde{x}| : xi in S}

where \tilde{x} = median of the data elements in S.

Excel Formula: If R is a range that 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 pressCtrl-Shft-Enter to obtain the array value, otherwise, the result wont come out correctly. This function only works properly when R doesnt contain any empty cell or cell with a non-numeric value.

Alternatively, you can use the 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 dont need to press Ctrl-Shft-Enter to use this function.

Example 8: IfS={2, 5, -1, 3, 4, 5, 0, 2}, the median absolute deviation = 2 sinceS ={-1, 0, 2, 2, 3, 4, 5, 5}, and so the median ofS = (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 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 rangeof 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 that 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 function RNG(R) which is contained in the Real Statistics Resource Pack.

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

Inter-quartile Range

Definition 7: The inter-quartilerange(IQR) of a data setSis 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 that contains the data elements inSthen the IQR of Scan be calculated in Excel by the formula:

=QUARTILE(R, 3) QUARTILE(R, 1)

Starting with Excel 2010 there is another version of the quartile function called QUARTILE.EXC. An alternative version of IQR is therefore

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

SeeRanking Functions in Excel for further information about the QUARTILE and QUARTILE.EXC functions. Alternatively, you can calculate the inter-quartile range via the functionIQR(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: IfS={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 Statisticsformulas =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: Thecoefficient of variation(aka the coefficient of variability), V (or CV), of the data setSis calculated as

V = s/x̄

Since s and x̄have the same units of measurement, Vhas 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 that contains the data elements inSthen the coefficient of variation forScan be calculated in Excel by the formula:

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

The population version of Vis σ/μ which can becalculated 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 seefrom 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 VA = .09/.12 = .75 and VB = .05/.08 = .625, stock B is considered to be the better investment since its relative risk (equal to its coefficient of variation) is lower.

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