Mean 

In statistics, the Mean is the average value of a given data, making it easier to compare and understand the data. It is an important measure of central tendency apart from Median and mode. In simple words, Mean is a value that is closer to all the values present in given data that we use to summarise that data. 

Mean Example: Imagine a set of scores for a class of students, ranging from high to low. To determine the overall performance of the class, you can calculate the mean, a single value that represents the average score that offers a clear understanding of how the class performed as a whole.

1.0Introduction to Mean

Mean Definition

The mean of a given set of data can be defined as the sum of all the values of data divided by the total number of values. It is also referred to as the Average of the data. The general mean formula is: 

$Mean=\frac{Sumofallvalues}{NumberofValues}$

Calculation of Mean

Calculating Mean for Ungrouped Data: 

The mean for ungrouped data is calculated as the sum of all individual data values divided by the total number of data points, giving a very simple average for the whole set. The mean formula for an ungrouped data say (x₁, x₂, x₃, x₄ …..,x_n) can be written as: 

$Mean=\frac{x_1+x_2+x_3+x_4\dots ..+x_n}{n}$

Calculating Mean for Grouped Data: 

Grouped data is the data that is arranged in intervals or classes. To calculate the mean of grouped data, you use the midpoint of each class, the class mark(x_i), and the frequency(f_i) of each class, multiply them together, and then divide by the total number of observations. Mathematically, the same can written as: 

$Mean(\bar{x})=\frac{\sum f_i{x}_i}{\sum f_i}$

2.0Other Values of Central Tendency

As mentioned earlier, the mean is not the only measure of central tendency. Median and Mode are the two other measures of central tendency equally important as the mean. 

1. Median 

The median helps identify the middle point of the data, making it useful in situations where the data is not symmetrically distributed. Unlike the mean, which is affected by large values, the median gives a more accurate measure of central tendency in these cases. 

How to find median for:

• For Ungrouped data: When there is an odd number of values, the median is the middle value, and when there is an even number, the median is the average of the two middle values.
• For Grouped Data: For Grouped data, the Median can be found by the following formula: 

$Median=L+(\frac{\frac{N}{2}-CF}{f})\times h$

Here:

• L = Lower boundary of the median class
• N = Total number of observations
• CF = Cumulative frequency of the class just before the median class
• f = Frequency of the median class
• h = Class width (difference between the upper and lower boundaries of the median class)

2. Mode: 

The mode is the specific value which occurs most frequently in a dataset. Mode is useful for identifying the most common value in a given dataset. To calculate the Mode: 

• For Ungrouped Data: It is the most number of times occurring value in a given data. 
• For grouped Data: The mode of grouped data can be calculated by the following formula: 

$Mode=L+(\frac{f_1-f_0}{2f_1-f_0-f_2})\times h$

Here, 

• L = Lower boundary of the modal class
• ​f₁ = Frequency of the modal class
• f₀  = Frequency of the class just before the modal class
• f₂  = Frequency of the class after the modal class
• h = Class width

3.0Relation Between Mean, Median and Mode

The relation between mean, median, and mode can be expressed using the following empirical formula: 

$Mode=3Median–2Mean$

4.0Practice Problems

Problem 1: Find the mean for the following data: 10, 12, 14, 15, 15, 18, 20, 20, 20, 25

Solution: For mean: 

The Sum of all the values = 10 + 12 + 14 + 15 + 15 + 18 + 20 + 20 + 20 + 25 = 169

Number of Values = 10

$Mean=\frac{Sumofallvalues}{numberofvalues}$

$Mean=\frac{169}{10}$

Mean=16.9

Problem 2: Find the mean, median, and mode for the following data: 5, 7, 8, 12, 15, 19, 23

Solution: 

For mean: 

$Mean=\frac{5+7+8+12+15+19+23}{7}$

$Mean=\frac{89}{7}$

Mean = 12.71

For Median: 

The number of values is odd n = 7 

So the median = $\frac{n+1}{2}=\frac{7+1}{2}=4^{th}$

Hence, the median of ungrouped data is 4_th term = 12

For Mode: 

The highest value of the given ungrouped data is 23, so it is the mode of the data.  

Problem 3: The following data represents the number of hours spent by students on homework per week (grouped data):

Find the mean, median, and mode. 

Solution: 

1. Mean

$Mean(\bar{x})=\frac{\sum f_i{x}_i}{\sum f_i}$

$Mean(\bar{x})=\frac{975}{39}=25$

2. Mode: 

For mode f₁ = 12 because it occurs the most times. So, f₀ = 8, f₂ = 10, h = 10, L = 20

$Mode=L+(\frac{f_1-f_0}{2f_1-f_0-f_2})\times h$

$Mode=20+(\frac{12-8}{2\times 12-8-10}\times 10$

$Mode=20+\frac{4}{6}\times 10$

$Mode=20+6.67=26.67$

3. Median: 

N = 39/2 = 19.5 

CF = 13 (since 19.5 is between 13 and 25)

L = 20, f = 12, h = 10 

$Median=L+(\frac{\frac{N}{2}-CF}{f})\times h$

$Median=20+(\frac{19.5-13}{12})\times 10$

$Median=20+(\frac{6.5}{12})\times 10$

$Median=20+5.417=25.417$