How to calculate the measures of central tendency, the mean, the mode and the standard deviation for ungroup data

This articles shows the simplest way of calculating the measures of central tendency, the mean, the mode and the standard deviation.

MEASURES OF CENTRAL TENDENCY

There are three main measures of central tendency. These include: the Mean, Mode and Median. 

THE MEAN:

The mean is the most important measure of central tendency. It is also called the arithmetic average. It measures the average of a set of number to determine the position of a number among the group of numbers. The mean can be computed for both grouped and ungrouped data.

In calculating the mean for ungrouped data, the following formula is used:

The equation above is pronounced as summation x divided by n where

Ʃ means the sum of

x stands for all the numbers in the distribution

n stands for the number of items in the distribution.

Example: calculate the arithmetic mean of the scores of the following students in a geography examination.

10      21      6        16      15     8        19      18      4        13

The first step to take is to add the whole numbers which will give us a total of 130 and this value is the summation of x

Then count the total numbers in the distribution which is 10

Therefore representing the above figures in our equation, we will have

 = 130

n = 10

 = 130/10 = 13

Therefore the arithmetic average for the ungrouped set of data representing the scores of some set of students in a geography examination is 13

Note: for the purpose of computing other measures of central tendency, it is advised that the array of data be arranged in an ascending order starting from the lowest to the highest number, although this is not compulsory when computing the mean.

COMPUTING THE MEDIAN FOR UNGROUPED DATA

The median is the value of the middle number in a distribution. In getting the median for ungrouped data, the set of data is arranged in an ascending order starting from the lowest to the highest number, and then the number at the middle becomes the median if and only if the number of items in distribution is odd. If the number if even as is the case in the numbers we used to compute the mean then add the two middle values and divide by two.

Example: compute the median for the following array of data.

10      21      6        16      15     8        19      18      4        13

Here to compute the median, we arrange to the numbers in ascending order starting from the lowest number

4        6        8        10      13      15      16      18      19      21

The two middle values in the distribution above are 13 and 15.

Therefore our median is 13+15 =28 then divide by 2 to get 28/2=14

The value of the median for the above scores is 14

COMPUTING THE MODE FOR UNGROUPED DATA

The mode is the value of the frequently occurred number in the distribution. If there are two numbers that have number the most in a distribution, it is called bi-modal. If is one it is called uni-modal.

The mode for the following grouped of data is

23      12      23      8        7        5        12      9        12      7

The mode for the above set of data is 12 because it appeared three times in the distribution unlike other ones that occurred just once or twice.

NOTE: we could not use the set of data we used for the mean and the median because it is a case of no mode. This is sometimes referred to as zero mode because no number appeared more than once.