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## Measure of Central Tendency or Measure of Central Location

We can define “Central Tendency” in three different ways.

A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. These measures indicate where most values in distribution fall and are also referred to as the central location of a distribution.

There are more than three measures of central tendency i.e. Mode, Median, Mean, Geometric Mean, Harmonic Mean etc. but first three are used most commonly.

## Mode

The mode is the most commonly occurring value in a data.

**Example**

Following is the number of problems that Mr. Zahid assigned for homework on 12 different days.

4, 7, 2, 8, 10, 14, 5, 8, 5, 8, 1, 12

To calculate the mode, the first step is to rearrange the data as:

1, 2, 4, 5, 5, 7, 8, 8, 8, 10, 12, 14

Here, the most commonly occurring value is 8. So, **mode** is **8**.

### Advantage of the Mode

The mode has an advantage over the median and the mean as it can be found for both numerical and categorical data.

### Limitations of the Mode

- In some distributions, the mode may not reflect the center of the distribution very well.

- The presence of more than one mode can limit the ability of the mode in describing the center or typical value of the distribution because a single value to describe the center cannot be identified.

- In some cases, particularly where the data are continuous, the distribution may have no mode at all i.e. all values are different.

## Median

The median is the middle value in distribution when the values are arranged in ascending or descending order. The median divides the distribution in half i.e. there are 50% of observations on either side of the median value.

- In a distribution with an odd number of observations, the median value is the middle value.

- In a distribution with an even number of observations, the median is the average of the two middle numbers.

**Example**

The arranged data, as we have supposed in an example while explaining mode is:

1, 2, 4, 5, 5, 7, 8, 8, 8, 10, 12, 14

To find out the middle number, we can use (n + 1)/2

Here n = 12 (the total quantities or numbers in a given data)

So, (12 + 1)/2 = 6.5

We need to find out the mean 6^{th} and 7^{th} value

(7 + 8)/2 = 7.5

So, **7.5** is the required **median**.

### Advantage of the Median

The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.

### Limitation of the Median

The median cannot be identified for categorical nominal data, as it cannot be logically ordered.

## Mean or Average

The mean is the sum of the value of each observation in a dataset divided by the number of observations. This is also known as the arithmetic average.

**Formula**

The formula of mean is usually written in a slightly different manner using the Greek capital letter, ∑ (“Sigma” which means “sum of”).

**Mean = ∑x / n**

**Example**

The arranged data, as we have supposed in an example while explaining mode is:

1, 2, 4, 5, 5, 7, 8, 8, 8, 10, 12, 14

Here, n = 12 and

∑x = 1 + 2 + 4 + 5 + 5 + 7 + 8 + 8 + 8 + 10 + 12 + 14

∑x = 84

Now, Mean = 84/12 = 7

So, **7** is the required **mean**.

### Advantage of the Mean

The mean can be used for both continuous and discrete numeric data.

### Limitations of the Mean

- The mean cannot be calculated for categorical data, as the values cannot be summed.

- As the mean includes every value in the distribution the mean is influenced by outliers and skewed distributions.

## When to Use Mode, Median And Mean?

To know what the best measure of central tendency, use the following summary table with respect to the different types of variables.