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Variance and standard deviation

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Unlike range and quartiles, the variance combines all the values in a data set to produce a measure of spread. The variance (symbolized by S2) and standard deviation (the square root of the variance, symbolized by S) are the most commonly used measures of spread.

We know that variance is a measure of how spread out a data set is. It is calculated as the average squared deviation of each number from the mean of a data set. For example, for the numbers 1, 2, and 3 the mean is 2 and the variance is 0.667.

[(1 - 2)2 + (2 - 2)2 + (3 - 2)2] ÷ 3 = 0.667

[squaring deviation from the mean] ÷ number of observations = variance

Variance (S2) = average squared deviation of values from mean

Calculating variance involves squaring deviations, so it does not have the same unit of measurement as the original observations. For example, lengths measured in metres (m) have a variance measured in metres squared (m2).

Taking the square root of the variance gives us the units used in the original scale and this is the standard deviation.

Standard deviation (S) = square root of the variance

Standard deviation is the measure of spread most commonly used in statistical practice when the mean is used to calculate central tendency. Thus, it measures spread around the mean. Because of its close links with the mean, standard deviation can be greatly affected if the mean gives a poor measure of central tendency.

Standard deviation is also influenced by outliers one value could contribute largely to the results of the standard deviation. In that sense, the standard deviation is a good indicator of the presence of outliers. This makes standard deviation a very useful measure of spread for symmetrical distributions with no outliers.

Standard deviation is also useful when comparing the spread of two separate data sets that have approximately the same mean. The data set with the smaller standard deviation has a narrower spread of measurements around the mean and therefore usually has comparatively fewer high or low values. An item selected at random from a data set whose standard deviation is low has a better chance of being close to the mean than an item from a data set whose standard deviation is higher.

Generally, the more widely spread the values are, the larger the standard deviation is. For example, imagine that we have to separate two different sets of exam results from a class of 30 students the first exam has marks ranging from 31% to 98%, the other ranges from 82% to 93%. Given these ranges, the standard deviation would be larger for the results of the first exam.

Standard deviation might be difficult to interpret in terms of how big it has to be in order to consider the data widely spread. The size of the mean value of the data set depends on the size of the standard deviation. When you are measuring something that is in the millions, having measures that are "close" to the mean value does not have the same meaning as when you are measuring the weight of two individuals. For example, a measure of two large companies with a difference of $10,000 in annual revenues is considered pretty close, while the measure of two individuals with a weight difference of 30 kilograms is considered far apart. This is why, in most situations, it is useful to assess the size of the standard deviation relative to the mean of the data set.

Although standard deviation is less susceptible to extreme values than the range, standard deviation is still more sensitive than the semi-quartile range. If the possibility of high values (outliers) presents itself, then the standard deviation should be supplemented by the semi-quartile range.

Properties of standard deviation

When using standard deviation keep in mind the following properties.

  • Standard deviation is only used to measure spread or dispersion around the mean of a data set.
  • Standard deviation is never negative.
  • Standard deviation is sensitive to outliers. A single outlier can raise the standard deviation and in turn, distort the picture of spread.
  • For data with approximately the same mean, the greater the spread, the greater the standard deviation.
  • If all values of a data set are the same, the standard deviation is zero (because each value is equal to the mean).

When analysing normally distributed data, standard deviation can be used in conjunction with the mean in order to calculate data intervals.

If x bar = mean, S = standard deviation and x = a value in the data set, then

  • about 68% of the data lie in the interval: mean - S < x < mean + S.
  • about 95% of the data lie in the interval: mean - 2S < x <mean + 2S.
  • about 99% of the data lie in the interval: mean - 3S < x < mean + 3S.

Discrete variables

The variance for a discrete variable made up of n observations is defined as:

Equation for the variance of a discrete variable.

The standard deviation for a discrete variable made up of n observations is the positive square root of the variance and is defined as:

Equation for the standard deviation of a discrete variable.

Use this step-by-step approach to find the standard deviation for a discrete variable.

  1. Calculate the mean.
  2. Subtract the mean from each observation.
  3. Square each of the resulting observations.
  4. Add these squared results together.
  5. Divide this total by the number of observations (variance, S2).
  6. Use the positive square root (standard deviation, S).

Example 1 – Standard deviation

A hen lays eight eggs. Each egg was weighed and recorded as follows:

60 g, 56 g, 61 g, 68 g, 51 g, 53 g, 69 g, 54 g.

  1. First, calculate the mean:
    Calculation of the mean for example 1 a.
  2. Now, find the standard deviation.

    Table 1.  Weight of eggs, in grams
    Weight (x) (x - mean) (x - mean)2
    60 1 1
    56 -3 9
    61 2 4
    68 9 81
    51 -8 64
    53 -6 36
    69 10 100
    54 -5 25
    472   320

    Using the information from the above table, we can see that

    The sum of each observation minus the mean, squared equals 320

    In order to calculate the standard deviation, we must use the following formula:

    Calculating the standard deviation for example 1 b.

Frequency table (discrete variables)

The formulas for variance and standard deviation change slightly if observations are grouped into a frequency table. Squared deviations are multiplied by each frequency's value, and then the total of these results is calculated.

In a frequency table, the variance for a discrete variable is defined as

Equation for variance of a discrete variable in a frequency table.

The standard deviation for a discrete variable is defined as

Equation for standard deviation of a discrete variable in a frequency table.

Example 2 – Standard deviation calculated using a frequency table

Thirty farmers were asked how many farm workers they hire during a typical harvest season. Their responses were:

4, 5, 6, 5, 3, 2, 8, 0, 4, 6, 7, 8, 4, 5, 7, 9, 8, 6, 7, 5, 5, 4, 2, 1, 9, 3, 3, 4, 6, 4

Table 2.  Thirty farmers were asked how many farm workers they hire during a typical harvest season. Their responses were:
Workers (x) Tally Frequency (f) (xf) (x - mean) (x - mean)2 (x - mean)2f
0 1 1 0 -5 25 25
1 1 1 1 -4 16 16
2 2 2 4 -3 9 18
3 3 3 9 -2 4 12
4 6 6 24 -1 1 6
5 5 5 25 0 0 0
6 4 4 24 1 1 4
7 3 3 21 2 4 12
8 3 3 24 3 9 27
9 2 2 18 4 16 32
    30 150     152

To calculate the mean:

Mean equals the sum of the data values times their frequency divided by the sum of the frequencies.

To calculate the standard deviation:

Calculation of standard deviation for example 2.

Example 3 – Standard deviation using grouped variables (continuous or discrete)

220 students were asked the number of hours per week they spent watching television. With this information, calculate the mean and standard deviation of hours spent watching television by the 220 students.

Table 3.  Number of hours per week spent watching television
Hours Number of students
10 to 14 2
15 to 19 12
20 to 24 23
25 to 29 60
30 to 34 77
35 to 39 38
40 to 44 8
  1. First, using the number of students as the frequency, find the midpoint of time intervals.
  2. Now calculate the mean using the midpoint (x) and the frequency (f).

Note: In this example, you are using a continuous variable that has been rounded to the nearest integer. The group of 10 to 14 is actually 9.5 to 14.499 (as the 9.5 would be rounded up to 10 and the 14.499 would be rounded down to 14). The interval has a length of 5 but the midpoint is 12 (9.5 + 2.5 = 12).

Calculation of the mean for example 3.

6,560 = (2 X 12 + 12 X 17 + 23 X 22 + 60 X 27 + 77 X 32 + 38 X 37 + 8 X 42)

Then, calculate the numbers for the xf, (x - mean), (x - mean)2 and (x - mean)2f formulas.

Add them to the frequency table below.

Table 4.  Number of hours spent watching television
Hours Midpoint (x) Frequency (f) xf (x - la moyenne) (x - la moyenne)2 (x - la moyenne)2f
10 to 14 12 2 24 -17.82 317.6 635.2
15 to 19 17 12 204 -12.82 164.4 1,972.8
20 to 24 22 23 506 -7.82 61.2 1,407.6
25 to 29 27 60 1,620 -2.82 8.0 480.0
30 to 34 32 77 2,464 2.18 4.8 369.6
35 to 39 37 38 1,406 7.18 51.6 1,960.8
40 to 44 42 8 336 12.18 148.4 1,187.2
    220 6,560      8,013.2

Example 4 – Standard deviation

Use the information found in the table above to find the standard deviation.

Calculation of the standard deviation using data from Table 4.

Note: During calculations, when a variable is grouped by class intervals, the midpoint of the interval is used in place of every other value in the interval. Thus, the spread of observations within each interval is ignored. This makes the standard deviation always less than the true value. It should, therefore, be regarded as an approximation.

Example 5 – Standard deviation

Assuming the frequency distribution is approximately normal, calculate the interval within which 95% of the previous example's observations would be expected to occur.

mean = 29.82, s = 6.03

Calculate the interval using the following formula: mean - 2s < x < mean + 2s

29.82 - (2 X 6.03) < x < 29.82 + (2 X 6.03)

29.82 - 12.06 < x < 29.82 + 12.06

17.76 < x < 41.88

This means that there is about a 95% certainty that a student will spend between 18 hours and 42 hours per week watching television.