Mean, Median and Mode

# Mean, Median and Mode

## Mean

The mean is the most common measure of central tendency used by the researchers. It is the measure of central tendency that is also referred to as the average. A researcher can use the mean to describe the data distribution of variables measured as intervals or ratios. These are variables that include numerically corresponding categories or ranges like race, class, gender, etc.

## How to Calculate the Mean

To calculate the mean of ungrouped data, we can use the following formulas:

1.      If the numbers of observations are less, i.e., less than 25, then we simply add all the observations and divide the sum by the total number of observations.

For example, if five families have 0, 2, 2, 3 and 5 children respectively, then the mean number of children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five families have an average of 2.4 children. We can use this formula to calculate the mean.
Here, xi = Sum of all the individual values and n= Total number of items.

2.      If the numbers of observations are more, i.e., more than 25, then we find the frequency (f) of distinct observations (x) and multiply them to find fx. Now, add all the values of fx to find fixiAgain, add all the frequencies to get fand use this formula to calculate the mean.

Example 1: The height of 10 boys is measured in centimeters, and the observations are as follows:
150, 139, 128, 152, 132, 146, 149, 141, 143, 135
Find the mean height.

Solution:
Mean height = Sum of all observations/Number of observations
= (128 + 132 + 135 + 139 + 141 + 143 + 146 + 149 + 150 + 152)/10
= 1415/10 cm = 141.5 cm
Thus, the mean height of the girls is 141.5 cm.

Example 2: The mean of four numbers 23, x, 17 and 29 is 20.5. Find the value of x.

Solution: Mean = (23 + x + 17 + 29)/4
20.5 = (69 + x)/4
69 + x = 82
x = 82 – 69
x = 13

## Median

The median is the value that occurs at the middle of a distribution of data when the data are organized in the ascending order. This measure of central tendency can be calculated for variables that are measured with ordinal, interval or ratio scales.
Calculation of the median is also very simple. Suppose we have the following list of numbers: 4, 9, 12, 41, 3, 67, 31, 7, 23. First, we must arrange the numbers in ascending order. The result is this: 3, 4, 7, 9, 12, 23, 31, 41, 67. The median is 12 because it is the exact middle number. There are four numbers before 12 and four numbers after 12.
If your data distribution has an even number of data values which means that there is no exact middle number, you simply adjust the data range slightly in order to calculate the median. For example, if we add the number 82 to the end of our list of numbers above, we have 3, 4, 7, 9, 12, 23, 31, 41, 67, 82, so there is no single middle number. In this case, take the average of the values for the two middle numbers. In our new list, the two middle numbers are 12 and 23. So, we take the average of those two numbers: (12 + 23) /2 = 17.5.

## How to Calculate Median

In individual series, where data is given in the raw form, the first step towards median calculation is to arrange the data in ascending or descending order. Now count the number of observations denoted by nThe next step is decided by whether the value of n is even or odd.
1.      If the value of n is odd, then

2.      If the value of n is even, then

Example 1: Find the median of the following.
24, 22, 18, 23, 26, 17, 28

Solution:
Arranging the data in ascending order: 17, 18, 22, 23, 24, 26, 28
Here n = 7, which is odd.
Median = Value of [(7 + 1)/2]th observation
= Value of 4th observation
Hence, median = 23

Example 2: Find the median of the following.
38, 42, 41, 40, 35, 37, 45, 36

Solution:
Arranging the data in ascending order: 35, 36, 37, 38, 40, 41, 42, 45
Here n = 8, which is even.
Median = Mean of (n/2)th and (n/2 +1)th observation
Median = Mean of 4th and 5th observation
Median = (38 + 40)/2 = 39

## Mode

The mode is the measure of central tendency that identifies the category or data value that occurs the most frequently within the distribution of data. In other words, it is the most common value or the score that appears the highest number of times in a distribution.
For example, suppose 100 families own the following pets:
• Dog: 80
• Cat: 25
• Duck: 11
• Parrot: 5
• Fish: 14
Here, the mode is "dog" since more families own a dog than any other animal. Note that the mode is always expressed as the category or score, not the frequency of that score. In the above example, the mode is "dog" not 80, which is the number of times dog appears.

## How to Calculate Mode

In case of raw data, we just have to count the data items and the item that occurs most frequently in the distribution is the mode of the series.

Example 1: Find the mode of the following data.
36, 48, 36, 44, 36, 42, 36, 48, 36, 40, 25, 42, 36, 44, 40

Solution:
The observation 36 has maximum frequency, i.e. 6.
Thus, mode = 36

Example 2: Find the mode of the following data.
14, 15, 13, 15, 16, 15, 13, 15, 16, 15, 13, 15, 16, 15, 13

Solution:
The observation 15 has maximum frequency, i.e. 7.
Thus, mode = 15