GMAT Exam: A Simple Guide To Arithmetic - Descriptive Measures of Data

Mean
The mean of a data set is the arithmetic average of the data values:


For example, the mean of 25, 43, 40, 60, and 12 is .

Median
The median is the middle value or the arithmetic average of the two middle values in an ordered set of data.

Fifty percent of the data values fall at or below the median and 50% fall at or above the median.

To find the median, put the data values in order from least to greatest (or greatest to least), and then locate the middle value.

If there is no single middle value, compute the average of the two middle values.

For example, after putting the data values in order, the median of 12, 25, 40, 43, 60 is 40. For –7, –7, 1, 8, 16, 22, the median is .

Tip: When finding a median, remember to put the numbers in order first.

In an ordered data set of n values, the median is in the position.

For example, in the dot plot shown, the median is in the position, meaning it is halfway between the 10th and 11th data values.



The 10th and 11th data values are both 10 minutes, so the median for the data shown in the dot plot is 10 minutes.

Mode
The mode is the data value (or values) that occurs with the greatest frequency in a data set.

For example, the mode of 30, 40, 50, 50, 60, and 70 is 50.

Tip: A data set in which each data value occurs the same number of times has no mode.

Range
The range gives the spread between the greatest and least values of the data set.

It is the difference between the maximum value and the minimum value in the data set. Thus, range = maximum value – minimum value.

For example, the range of 6.7, 7.6, 7.5, 6.9, 9.3, 6.7, 7.6, and 8.5 is 9.3 – 6.7 = 2.6.

Mean Absolute Deviation
The mean absolute deviation (MAD) quantifies the degree to which the data values vary from their mean.

It is the average distance between each data value and the mean of the data values.

The MAD for 30, 40, 50, 60, and 70 (which has a mean of 50) is



Variance and Standard Deviation
Like the MAD, the variance and standard deviation are measures of the variability of a set of data values about the mean.

If there is no variability in a data set, each data value equals the mean, so both the variance and standard deviation for the data set are zero.

The more the data values vary from the mean, the greater are the variance and standard deviation.

You compute the variance by subtracting the mean from each data value, squaring each difference, and then summing up the squared differences and dividing by the number, n, of data values.

The standard deviation is the square root of the variance.

Note: When the data set is a sample from a population, you divide by n – 1 instead of n.

Questions about variance or standard deviation on the GMAT might require you to examine two (or more) data sets with equal means and decide which has the greater (or lesser) variance or standard deviation.

For example, the data set 30, 40, 50, 60, and 70 (which has a mean of 50) has a greater variance and standard deviation than does the data set 35, 47, 50, 53, 65 (which also has a mean of 50). The data in the second data set are clustered more tightly around the mean.