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# Central Tendency Formula

Before learning the central tendency formula, let us recall what is central tendency. A measure of central tendency describes a set of data by identifying the central position in the data set as a single value. We can think of it as a tendency of data to cluster around a middle value. Let us learn the central tendency formulas in the upcoming section.

## What Is the Central Tendency Formula?

In statistics, the three most common measures of central tendencies are mean, median, and mode. Here are the central tendency formulas for each of them.

**Formula to Calculate Mean**

The mean formula of given observations can be expressed as,

Mean = Sum of Observations ÷ Total Number of Observations

**Formula to Calculate Median**

The Median Formula of a given set of numbers, say having 'n' observations, can be expressed as:

If 'n' is odd:

Median = (n + 1)/2 ^{th }term

If 'n' is even:

Median = \( \dfrac{ \frac{n}{2}^{\text{th}} \text{term} + (\frac{n}{2} + 1)^{\text{th}} \text{term}}{2}\)

**Formula to Calculate Mode**

Mode for grouped data is found using the following formula.

\(\begin{align}Mode = L + h\dfrac{(f_m-f_1)}{(f_m-f_1)-(f_m-f_2)}\end{align}\)

Where,

- '\(\begin{align}L\end{align}\)' is the lower limit of the modal class
- '\(\begin{align}h\end{align}\)' is the size of the class interval
- '\(\begin{align}f_m\end{align}\)' is the frequency of the modal class
- '\(\begin{align}f_1\end{align}\)' is the frequency of the class preceding the modal class
- '\(\begin{align}f_2\end{align}\)' is the frequency of the class succeeding the modal class

Let us see the applications of the central tendency formulas in the following section.

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## Solved Examples Using Central Tendency Formula

**Example 1:** Age of students = 13, 14, 16, 13, 18, 20, 24, 13. Find the mode.

**Solution:**

Since there is only one value repeating itself, it is a unimodal list (by one of the central tendency formulas).

Mode = 13

**Answer: **Mode = 13

**Example 2:** The measurements of heights of people in the club are listed below: { 41, 33, 36, 29, 40, 38, 42}. Using the median formula, calculate the median of the above-given data.

**Solution:**

To find the median of the given set.

Given:

Set of ages for different members: {42, 38, 29, 36, 40, 33, 41}

Arranging the set in ascending order: {29, 33, 36, 38, 40, 41, 42}

Number of observations, n = 7 (odd)

Using median formula (one of the central tendency formulas).,

Median = (7 + 1)/2 ^{th }term

= 4^{th }term

= 38

**Answer: **Median of the given set = 38

**Example 3: **What will be the mean of the first five natural even numbers?

**Solution :**

Given data is 2, 4, 6, 8,10.

Using Mean Formula,

Mean=(2+4+6+8+10) ÷ 5

= 30 ÷5 = 6

**Answer:** 6 is the mean of the first five natural even numbers.

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