# Chapter 2 : Measure of Central Tendency

### Topics covered in this snack-sized chapter:

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.
It can be measured by:
Variation can be measured by:
It is the most common measure of central tendency.
The mean is equal to the sum of all the values in the data set divided by the number of values in the data set.

Example: Find the mean of the following:
{66, 72, 83, 89}
- Mean = (66 + 72 + 83 + 89) / 4

###### Sample Mean :

If n is the sample size then we have:

Where,
- Sum of all values =

- If the data represents a sample, the number of entries = n

###### Population Mean :

If N is population size then we have:

Where,
Sum

Sum of all values of

- If the data represents an entire population, the number of entries =N

It is used when the only source of data is a frequency distribution.

Where,
n = Sample size,

c = Number of classes in the frequency distribution,

m_{j
}= Midpoint of the j^{th
}^{
}class,

f_{j
}= Frequency of the j^{th
} class

The median is the middle score for a set of data that has been arranged in ascending or descending order.

##### Step: 1

Arrange numbers in ascending order.
##### Step: 2A

For odd numbers:

##### Step: 2B

For even numbers median is the average of the (n/2)^{th
} observation and the observation.
Example 1: Find the median of the following:
72, 65, 81, 89, 83
- Solution: Arrange the numbers in ascending order.

65, 72, 81, 83, 89

For odd numbers:

- Here the median is 3
^{rd
} observation that is 81.

Example 2: Find the median of the following:
Solution: Arrange the numbers in ascending order.
12, 22, 31, 43, 50, 57
For even numbers:
- The median is the average of the middle two values,

Here median is 37.
Outliers are numbers in a data set that are either way bigger or way smaller than the other numbers in a data set.
Example: In 1, 2, 3, 4, 4, 6 and 31 data set the number 31 is the outlier.

###### Mean is impacted by Outliers:

If the outlier is a high value, it will cause the mean value to shift to the higher side, while a low valued outlier will drop the mean value to a lower number.
Below example is showing how the mean is impacted by outliers:

###### Median is not impacted by Outliers:

Median is not impacted by outliers as shown below:

Mode is a number that occurs most frequently in the data set.
__Steps to find Mode:
__

Step: 1
- Arrange the numbers in ascending or descending order.

Step: 2
- Find the number which is occurring maximum number of times in the set.

##### Note:

If there are two such numbers which occur maximum number of times then there is NO mode.
Example 1: Find the mode of the following:
{9, 3, 3, 7, 8, 15, 3, 9}

- Solution: Arrange the numbers in ascending order:

- As 3 is occurring maximum number of times, 3 is the mode.

Example 2: For data:
6, 7, 2, 5, 3, 4, 9, 8

- As no item in the data set given above is repeating itself so there will be No mode.

The difference between the largest and the smallest values of a distribution is known as range.
Example 1: The range of 10, 13, 17, 17, 18 will be:

Percentile is the value of a variable below which a certain percent of observations fall.
Example: The 30^{th
} percentile is the value below which 30 percent of the observations may be found.
The percent falling above the percentile P^{th
} will be (100 – P)%.

Quartiles split ordered data into 4 equal portions.
Each Quartile has position and value.

With the data in an ordered array, the position of is:

The value of is the value associated with that position in the ordered array.
It is also known as mid-spread or middle fifty.
It is equal to the difference between upper quartile Q_{1
}and lower quartile Q_{3
}.
- The middle is 50% of the values.

- Resistant to extreme values.

Interquartile range = IQR = Q_{3
} - Q_{1
}