Class 10 Maths Statistics | Central Tendency |

**Central Tendency**

** **It is a measure that tells us where the middle of a bunch of data lies.

3 most common measures of central tendency are the **mean**, the **median**, and the **mode**.

Let’s take a situation where Mary & Hari got marks for 5 questions. Both of them claim to have performed better. Let’s see who has performed better in the exam.

**Mean**: Let’s find the average scores.

- Average Score of Mary is (10+8+9+8+7)/5 =8.4
- Average Score of Hari is (4+7+10+10+10)/5 = 8.2

Based on the Mean data, Mary has performed better.

The **mean **(or **average**) of a number of observations is the sum of the values of all the observations divided by the total number of observations. It is denoted by ‘**x **bar’.

All data is given equal importance in Mean, the extreme values are not ignored.

For Ungrouped data,

**Median**: Let’s arrange their score in ascending order in blue table. Now we will take the middle most score. Since we are talking about central tendency of data, middle most score should also reflect central tendency of data. Middle most score for May is 8, while that of Hari is 10.

So based on middle score or median data, Hari has performed better.

When the number of observations (n) is odd, the median is the value of the ((n+1)/2) ^{Th} observation.

When the number of observations (n) is even, the median is the mean of (n/2) & (n/2 + 1)^{ Th} observations.

Median Ignores extreme values & Tells the point from where 50% data is lesser and 50% data is more

**Mode**: Let’s check the most frequent score. Based on the most often score, 8 is the most often score( 2 times) for Mary & 10 is the most often score ( 3 times) for Hari. Going with the most often score or Mode, Hari has performed better.

The mode is that value of the observation which occurs most frequently, i.e., an observation with the maximum frequency is called the mode. E.g. it can be used to find the which is the most favorite subject of a class.

**Numerical:** In a mathematics test given to 15 students, following marks are recorded: 41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60. Find the mean, median and mode of this data.

Mean is average data, Median is middle most data & Mode is the most frequent data.

Mean = Sum of all observations / number of observations

= (41+ 39+ 48+ 52+ 46+ 62+ 54+ 40+ 96+ 52+ 98+ 40+ 42+ 52+ 60 ) / 15

= 822/15

= 54.8

To find middle most data, we have to arrange data in ascending order & then find middle most.

Data in ascending order: 39, 40, 40, 41, 42, 46, 48, * 52*, 52, 52, 54, 60, 62, 96, 98.

Middlemost data in this case is 52 & thus Median is 52.

Also we see that most frequent data is 52 & thus mode is 52.

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