13.3: Measures of Spread

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Learning Objectives

By the end of this section, you will be able to:

Define and calculate standard deviation for a data set.
Define and calculate variance for a data set.
Explain the relationship between standard deviation and variance.

Standard Deviation

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated close to the mean; in other data sets, the data values are more widely spread out. For example, an investor might examine the yearly returns for Stock A, which are 1%, 2%, -1%, 0%, and 3%, and compare them to the yearly returns for Stock B, which are -9%, 2%, 15%, -5%, and 0%.

Notice that Stock B exhibits more volatility in yearly returns than Stock A. The investor may want to quantify this variation in order to make the best investment decisions for a particular investment objective.

The most common measure of variation, or spread, is standard deviation. The standard deviation of a data set is a measure of how far the data values are from their mean. A standard deviation

provides a numerical measure of the overall amount of variation in a data set; and
can be used to determine whether a particular data value is close to or far from the mean.

The standard deviation provides a measure of the overall variation in a data set. The standard deviation is always positive or zero. It is small when the data values are all concentrated close to the mean, exhibiting little variation or spread. It is larger when the data values are more spread out from the mean, exhibiting more variation.

Suppose that we are studying the variability of two different stocks, Stock A and Stock B. The average stock price for both stocks is $5. For Stock A, the standard deviation of the stock price is 2, whereas the standard deviation for Stock B is 4. Because Stock B has a higher standard deviation, we know that there is more variation in the stock price for Stock B than in the price for Stock A.

There are two different formulas for calculating standard deviation. Which formula to use depends on whether the data represents a sample or a population. The notation s is used to represent the sample standard deviation, and the notation $σ$ is used to represent the population standard deviation. In the formulas shown below, x̄ is the sample mean, $μ$ is the population mean, n is the sample size, and N is the population size.

Formula for the sample standard deviation:

s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n - 1}}

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Formula for the population standard deviation:

σ = \sqrt{\frac{\sum (x - μ)^{2}}{N}}

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Variance

Variance also provides a measure of the spread of data values. The variance of a data set measures the extent to which each data value differs from the mean. The more the individual data values differ from the mean, the larger the variance. Both the standard deviation and the variance provide similar information.

In a finance application, variance can be used to determine the volatility of an investment and therefore to help guide financial decisions. For example, a more cautious investor might opt for investments with low volatility.

Similar to standard deviation, the formula used to calculate variance also depends on whether the data is collected from a sample or a population. The notation $s^{2}$ is used to represent the sample variance, and the notation σ² is used to represent the population variance.

Formula for the sample variance:

s^{2} = \frac{\sum (x - \bar{x})^{2}}{n -} 1

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Formula for the population variance:

σ^{2} = \frac{\sum (x - μ)^{2}}{N}

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This is the method to calculate standard deviation and variance for a sample:

First, find the mean $\bar{x}$ of the data set by adding the data values and dividing the sum by the number of data values.
Set up a table with three columns, and in the first column, list the data values in the data set.
For each row, subtract the mean from the data value $(x - \bar{x})$ , and enter the difference in the second column. Note that the values in this column may be positive or negative. The sum of the values in this column will be zero.
In the third column, for each row, square the value in the second column. So this third column will contain the quantity (Data Value – Mean)² for each row. We can write this quantity as ${(x - \bar{x})}^{2}$ . Note that the values in this third column will always be positive because they represent a squared quantity.
Add up all the values in the third column. This sum can be written as $\sum {(x - \bar{x})}^{2}$ .
Divide this sum by the quantity (n – 1), where n is the number of data points. We can write this as $\frac{\sum {(x - \bar{x})}^{2}}{n - 1}$ .
This result is called the sample variance, denoted by s². Thus, the formula for the sample variance is $s^{2} = \frac{\sum {(x - \bar{x})}^{2}}{n - 1}$ .
Now take the square root of the sample variance. This value is the sample standard deviation, called s. Thus, the formula for the sample standard deviation is $s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n - 1}}$ .
Round-off rule: The sample variance and sample standard deviation are typically rounded to one more decimal place than the data values themselves.

Think It Through

Finding Standard Deviation and Variance

A brokerage firm advertises a new financial analyst position and receives 210 applications. The ages of a sample of 10 applicants for the position are as follows:

40, 36, 44, 51, 54, 55, 39, 47, 44, 50

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The brokerage firm is interested in determining the standard deviation and variance for this sample of 10 ages.

Solution

Find the sample variance and sample standard deviation by creating a table with three columns (see Table 13.3).

The data set is 40, 36, 44, 51, 54, 55, 39, 47, 44, 50.
This data set has 10 data values. Thus, $n = 10$ .
The mean is calculated as $\bar{x} = 46$ .
Column 1 will contain the data values themselves.
Column 2 will contain $x - \bar{x}$ .

Column 3 will contain

{(x - \bar{x})}^{2}

Table 13.3: Calculations for Standard Deviation for Age Example
Column 1 $x$	Column 2 $(x - \bar{x})$	Column 3 ${(x - \bar{x})}^{2}$
40	$40 - 46 = - 6$	$(- 6)^{2} = 36$
36	$36 - 46 = - 10$	$(- 10)^{2} = 100$
44	$44 - 46 = - 2$	$(- 2)^{2} = 4$
51	$51 - 46 = 5$	$(5)^{2} = 25$
54	$54 - 46 = 8$	$(8)^{2} = 64$
55	$55 - 46 = 9$	$(9)^{2} = 81$
39	$39 - 46 = - 7$	$(- 7)^{2} = 49$
47	$47 - 46 = 1$	$(1)^{2} = 1$
44	$44 - 46 = - 2$	$(- 2)^{2} = 4$
50	$50 - 46 = 4$	$(4)^{2} = 16$
	$Sum = 0$	$Sum = 380$

To calculate the sample variance, use the sample variance formula:
$s^{2} = \frac{\sum {(x - \bar{x})}^{2}}{n - 1} = \frac{380}{10 - 1} = \frac{380}{9} \approx 42.2$
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