Mean in Statistics

Mean (in statistics) is the average of a set of numbers. It is one of the most important measures of central tendency in distributed data.

• To calculate the mean, add all the values in the data set and then divide the total by the number of values.
• The mean is usually denoted by x̄ (read as “x bar”).

Formula

The mean formula in statistics is defined as the sum of all observations in the given dataset divided by the total number of observations.

Example: Calculate the mean of the first 10 natural numbers.

Solution:

First 10 natural numbers = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Sum of first 10 natural numbers = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
Mean = Sum of 10 natural numbers/10
⇒ Mean = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)/10
⇒ Mean = 55/10
⇒ Mean = 5.5

How to Find the Mean?

To find the mean of a dataset, it's important to first determine whether the data is grouped or ungrouped, as the method of calculation differs for each.

Mean of Ungrouped Data

Ungrouped data is raw data that is not organized into any groups or intervals. These are individual observations listed as they are.

For ungrouped data, the mean is found by adding all individual values and dividing by the total number of observations.

Formula:

Example: Heights of 10 students: 142, 145, 150, 152, 148, 149, 151, 147, 146, 150

• Sum of heights = 1,480
• Number of students = 10

xˉ = 1480/10 = 148 cm

So, the mean height of the 10 students is 148 cm.

Mean with Frequency Distribution

Let's assume there are n number of items in a set, i.e., {x₁, x₂, x₃, ... x_n}, and the frequency of each item is given as {f₁, f₂, f₃, . . ., f_n}. Then, the mean is calculated using the formula:

OR

Mean of Grouped Data

Grouped data is data that has been organized into groups (classes or intervals) to make it easier to understand and analyze, especially when the data set is large.

To represent grouped data, we use a frequency distribution table, which shows how many observations fall into each interval.

The mean of grouped data can be calculated using three methods:

• Direct Method: Mean of grouped data using class marks and their corresponding frequencies.
• Assumed Mean Method: Simplifies calculations by taking a convenient value as the assumed mean.
• Step Deviation Method: Further simplifies calculations by using scaled deviations of class marks.

Types of Mean

In statistics, there are four types of mean:

• Arithmetic Mean: Obtained by dividing the sum of all observed values by the total number of observations.
• Geometric Mean: The nth root of the product of all values in the dataset.
• Harmonic Mean: Calculated by dividing the number of observations by the sum of the reciprocals of the values.
• Weighted Mean: Calculated by assigning weights to values according to their importance and then finding the average.

Note: When not specified, the mean is generally referred to as the arithmetic mean.

Solved Questions

Question 1: A man keeps a record of the number of steps he jogs each day throughout the week. His step count for each day is recorded as follows:

• Monday: 8000
• Tuesday: 7500
• Wednesday: 8200
• Thursday: 7900
• Friday: 8100
• Saturday: 7800
• Sunday: 7700

Using this information, calculate the mean (average) number of steps he jogged per day

Solution:

The mean number of steps is shown in the graph below.

👁 mean_step_count_over_7_days

Sum of steps = 8000 + 7500 + 8200 + 7900 + 8100 + 7800 + 7700 = 54,200

Mean = 54,200 ÷ 7 = 7,886 steps

Question 2: Calculate the mean of the first 5 even natural numbers.

Solution: 

Given,

• Observed first 5 even natural numbers 2, 4, 6, 8, 10
• Total number of observed values = 5

Using Mean Formula

Mean = (Sum of observed values in data)/(Total number of observed values in data)

⇒ Sum of observed values = 2 + 4 + 6 + 8 + 10 = 30

Total number of observed values = 5
⇒ Mean = 30/5
⇒ Mean = 6 

Therefore, mean for first 5 even numbers = 6

Question 3: Calculate the mean of the first 10 natural odd numbers.

Solution: 

Given,

• Observed first 5 odd natural numbers 1, 3, 5, 7, 9.
• Total number of observed values = 5

Using Mean Formula

Mean = (Sum of observed values in data)/(Total number of observed values in data)

Sum of observed values = 1 + 3 + 5 + 7 + 9 = 25

Total number of observed values = 5

⇒ Mean = 25 / 5
⇒ Mean = 5

Therefore, mean for first 5 odd numbers = 5

Question 4: Calculate missing values from the observed set 2, 6, 7, x, whose mean is 6.

Solution:

Given,

• Observed values 2, 6, 7, x
• Number of observed values = 4
• Mean = 6

Using Mean Formula

Mean = (Sum of observed values in data)/(Total number of observed values in data)

⇒ Sum of observed values = 2 + 6 + 7 + x = 15 + x

Total number of observed values = 4

⇒ 6 = (15 + x)/4
⇒ 6 × 4 = 15 + x
⇒ x = 9

Therefore, missing value from the set is 9

Question 5: There are 20 students in Class 10. The marks obtained by the students in mathematics (out of 100) are given below. Calculate the mean of the marks.

Solution:

Given,

• Total number of students in class 10 = 20
• x₁ = 100, x₂ = 92, x₃ = 80, x₄ = 75, x₅ = 70
• f₁ = 1, f₂ = 3, f₃ = 5, f₄ = 10, f₅ = 1

Using Mean Formula

⇒ x̄ = {(100 × 1) + (92 × 3) + (80 × 5) + (75 × 10) + (70 × 1)}/20
⇒ x̄ = (100 + 276 + 400 + 750 + 70)/20 
⇒ x̄ = 1596/20 = 79.8 marks

Question 6: Calculate the mean of the following dataset.

Solution:

Range of data is 60 to 76, for assumption of mean, lets take average of the range values,

Assumed Mean = (60 + 76) /2 = 136/2 = 68

Now, Let's A = 68 be assumed mean of the data,

Now, using assumed mean value, let's create the table for step deviation as follows:

Height (in inches)	Frequency(fi)	Class Mark (xi)	Deviation (di)	Step Deviation (ui)	fi × ui
60 - 62	2	61	-7	-3.5	-7
62 - 64	3	63	-5	-2.5	-7.5
64 - 66	4	65	-3	-1.5	-6
66 - 68	6	67	-1	-0.5	-3
68 - 70	5	69	1	0.5	2.5
70 - 72	3	71	3	1.5	4.5
72 - 74	1	73	5	2.5	2.5
74 - 76	1	75	7	3.5	3.5
∑f = 25	∑fiui = -10.5

Thus, Mean = 68 + 2 × (-10.5)/25 
⇒ Mean = 68 + 2 × (-0.42) 
⇒ Mean = 68 - 0.84 = 67.16

Thus, mean height of data using step deviation method is 67.16 inches.

Thus, Mean = 68 + 2 × (-10.5)/25 

⇒ Mean = 68 + 2 × (-0.42) 

⇒ Mean = 68 - 0.84 = 67.16

Thus, the mean height of the data using the step deviation method is 67.16 inches.

Question 7 : Heights of 100 students grouped into intervals:

Solution:

• Total frequency:

∑f_i = 12 + 28 + 35 + 25 = 100

• Sum of f_ix_i:

1710 + 4144 + 5355 + 3950 = 15159

xˉ = 15159/100 ​= 151.59 cm

So, the mean height of the 100 students is 151.59 cm.

Practice Questions

Question 1: Find the Mean temperature of a week given that the temperatures from Monday to Sunday are 21℃, 23℃, 22.5℃, 21.6℃, 22.3℃, 24℃, 20.5℃.

Question 2: Find the mean of the first 10 even numbers.

Question 3: Find the Mean height of students if the given heights are 150 cm, 152 cm, 155 cm, 160 cm, and 148 cm.

Question 4: Find the Mean of the given dataset

Related Articles

• Arithmetic vs Geometric Mean