__maths__>__Statistics and Probability__>__Statistics : Analysis of Data__### Representative Values of Data : Mean, Median, Mode

In this page, the mean, median, mode of a data is introduced with examples.

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We have studied about representation of data in

tally marks

tabular form

pictograph

bar-graph

What is common between these representations?

- each of these helps to visualize and understand the data
- each of these helps to visualize and understand the data
- these do not help to understand the data

The answer is "each of these helps to visualize and understand the data"

Consider the data : The height values of five students from a class are given below in centimeters.

`90`, `97`, `91`, `92`, `90`.

All these students are of the same age. Another student of the same age is joining the class. Which of the following helps in predicting how tall the new student would be?

- cannot predict the height of a person, unless you see them.
- the new student will be about the same height as the current students, little more or less.
- the new student will be about the same height as the current students, little more or less.

The answer is "the new student will be about the same height as the current students, little more or less.".

The height values of five students from a class are given below in centimeters.

`90`, `97`, `91`, `92`, `90`.

This data is used to predict the height of student of the same age. Instead of giving the entire data, can the data be summarized? To summarize the data, there are three possible solutions.

1. Mean: The value when all data are evenly distributed among the given data values. This involves addition and division.

2. Median: The value that is in the middle of the increasing or decreasing order of the given data values. This involves arranging in increasing or decreasing order.

3. Mode: The data value that is repeating most number of times in the given data values. This involves counting.

Each of these three is explained with the given dataset.

The height values of `color(coral)(5)` students from a class are given below in centimeters.

`color(deepskyblue)(90)`, `color(deepskyblue)(97)`, `color(deepskyblue)(91)`, `color(deepskyblue)(92)`, `color(deepskyblue)(90)`.

The given values are added and divided by the number of values

mean

`=color(deepskyblue)((90+97+91+92+90))//color(coral)5`

`=color(deepskyblue)(460)//color(coral)(5)`

`=92` This is understood as "evenly distributing the values". This is illustrated in the figure.

What does the calculation show?

- the mean of the given data is `92`
- even distribution results in all values `92`
- both the above
- both the above

The answer is "both the above"

Which of the following is a meaning for the word "mean"?

- showing high intensity and eager enjoyment
- value of each data point when all are grouped together and evenly distributed among them
- value of each data point when all are grouped together and evenly distributed among them

The answer is "value of each data point when all are grouped together and evenly distributed among them".

What is the term used to refer "value of each data point when all are grouped together and evenly distributed among them"?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is "mean".

Mean is also called arithmetic mean.

Mean is also called as average by some authors. * Generally, average includes all three measures mean, median, and mode. These three measures are also called as central tendencies. *

The height values of `color(coral)(5)` students from a class are given below in centimeters.

`color(deepskyblue)(90)`, `color(deepskyblue)(97)`, `color(deepskyblue)(91)`, `color(deepskyblue)(92)`, `color(deepskyblue)(90)`.

The given values are arranged in increasing order and middle one is found.

median

`=text( middle of increasing order )color(deepskyblue)((90, 97, 91, 92, 90))`

`=text( middle value of )color(deepskyblue)((90, 90, 91, 92, 97))`

`=91` This is understood as "removing equal number of data values that are smaller and larger". This is illustrated in the figure.

What does the calculation show?

- the median of the given data is `91`
- removing equal number of smaller and larger value results in the value `91`
- both the above
- both the above

The answer is "both the above"

Which of the following is a meaning for the word "median"?

- a wall built to split two parts of an agricultural field
- the data value at the middle with equal number of smaller and larger data values
- the data value at the middle with equal number of smaller and larger data values

The answer is "the data value at the middle with equal number of smaller and larger data values".

What is the term used to refer "the data value at the middle with equal number of smaller and larger data values"?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is "median".

The height values of `color(coral)(5)` students from a class are given below in centimeters.

`color(deepskyblue)(90)`, `color(deepskyblue)(97)`, `color(deepskyblue)(91)`, `color(deepskyblue)(92)`, `color(deepskyblue)(90)`.

The values that is repeating most number of times is found.

`90` is repeated twice, and other data points `91`, `92`, `97` are repeated only once. So,

mode

`=90` This is understood as "the value that repeats most number of times". This is illustrated in the figure.

What does the calculation show?

- the mode of the given data is `90`
- the value that repeats the most number of times is `90`
- both the above
- both the above

The answer is "both the above"

Which of the following is a meaning for the word "mode"?

- most repeating data value
- most repeating data value
- method with which lowest value is found

The answer is "most repeating data value".

What is the term used to refer "most repeating data value"?

- Pronunciation : Say the answer once

Spelling: Write the answer once

The answer is "mode".

**Mean** : The value when the data is evenly distributed to all the values.

**Simplified Procedure to calculate Mean** : Add all the data values and divide the sum by the number of data-points.

**Median** : The value that has equal number of smaller and larger data values in the given data.

**Simplified Procedure to calculate Median** : Arrange the data values in ascending order and find the value in the middle.

**Mode** : The value that is repeated most number of times in the data.

**Simplified Procedure to calculate Mode** : Count the number of times a data-point repeats and find the data value with most repetition.

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