Descriptive Measures

Measures of central tendency and dispersion are common descriptive measures for summarising numerical data.

1. Measures of central tendency:

Measures of central tendency are measures of the location of the middle or the center of a distribution.

The most frequently used measures of central tendency are the mean, median and mode.

The mean is obtained by summing the values of all the observations and dividing by the number of observations.
The median (also referred to as the 50th percentile) is the middle value in a sample of ordered values. Half the values are above the median and half are below the median.
The mode is a value occurring most frequently. It is rarely of any practical use for numerical data.

A comparison of the mean, median and mode can reveal information about skewness, as illustrated in figure below. The mean, median and mode are similar when the distribution is symmetrical. When the distribution is skewed the median is more appropriate as a measure of central tendency.

2. Measures of Dispersion:

A measure of dispersion is a numerical value describing the amount of variability present in a data set.

The standard deviation (SD) is the most commonly used measure of dispersion. With the SD you can measure dispersion relative to the scatter of the values about their mean.

The range can also be used to describe the variability in a set of data and is defined as the difference between the maximum and minimum values. The range is an appropriate measure of dispersion when the distribution is skewed.

Example:

Consider the following measurements: 8, 9, 11, 5, 12, 17, 7, 23, 39, 15 . (Click here for information on how to use the calculator)

a) Calculate the measures of central tendency: Mean, Median and Mode:

The mean is 14.6
To locate the median, data values are ordered: 5, 7, 8, 9, 11, 12, 15, 17, 23, 39. The median lies between the 5th and 6th value = 11.5.
Mode: Not applicable, since each value occures only once.

b) Calculate the measures of dispersion: Range and Standard Deviation

Range: 5 to 39.
Standard deviation: 10.09

Interactive Examples

The Examples listed below have been included in order to illustrate the concepts discussed in this Section.

Question 1	Question 6	Question 11
Question 2	Question 7	Question 12
Question 3	Question 8	Question 13
Question 4	Question 9	Question 14
Question 5	Question 10	Question 15

How to use the calculator:

Click on calculator: (Alternately, if you have a Microsoft-compatible keyboard, press the "Calculator" button on the keyboard.)
You may have to set the calculator to scientific mode by clicking on the VIEW tab as it might be set to standard mode.
Then do the following to enter the data in order to compute the mean and the standard deviation:
1. Enter your first data value.
2. Click Sta, and then click Dat.
3. Enter the rest of the data, clicking Dat after each entry.
4. Click the Ave button to calculate the mean, click the s button to calculate the standard deviation.

BS1.4: Descriptive Measures

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