Mean in Statistics

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Mean in Statistics

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In statistics, three measures are defined as central tendencies that are: Mean, Median, and Mode, where the mean provides the average value of the dataset, the median provides the central value of the dataset, and the most frequent value in the dataset is the mode.

Calculation of central tendency, such as mean, median, and mode, is very useful in a lot of fields of study, such as Data Science, Statistics, and Machine Learning.

In this article, we’ll explore everything you need to know about the mean, including its formula, examples, the mean of grouped and ungrouped data, and more in detail.

Table of Content

Mean

Mean in Mathematics is the measure of central tendency and is mostly used in Statistics. Mean is the easiest of all the measures. The method of finding the mean is also different depending on the type of data. Data is of two types, grouped data and ungrouped data. The mean is generally the average of a given set of numbers or data. It is one of the most important measures of the central tendency of distributed data.

It is calculated by adding all the numbers in the data set and dividing by the number of values in the set. The mean is also known as the average. It is sensitive to skewed data and extreme values. For example, when the data is skewed, it can miss the mark.

For example, a man keeps a record of the number of steps he jogs daily throughout the week. The mean number of steps is shown in the graph below.

mean_step_count_over_7_days — Average of number of steps in a week

Mean Symbol

Mean is denoted as a bar over x or \bar{x}. Let's say the dataset given is X = {x₁, x₂, x₃,..., x_n} The mean of this dataset is denoted as μ \bar{x}and is given by:

\bar{x} = \frac{x_1 + x_2 + x_3 +...x_n}{n}

Mean Application

There are many uses and examples of the mean in real life. The following are some of the real-life examples of mean:

The average (mean) marks obtained by the students in a class.
A cricketer's average is also an example of a mean.
The average salary package is also used for the marketing of the colleges and their placement cell.

Mean 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. The image added below shows the mean formula of the given observation.

We use a mean formula to easily calculate the mean of a given dataset set for example,

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.

For ungrouped data (individual data points listed without any frequency distribution), the mean is calculated by summing all the values and dividing by the number of observations.
For grouped data (data presented in class intervals with frequencies), a different formula is used that incorporates class midpoints and frequencies.

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

There are two steps involved in the calculation of the mean:

Step 1: Calculate the sum of observed values in the data.
Step 2: Divide the sum of observed values into the number of observed values in the data.

Based on the type of dataset given, we can find out the mean using different methods. Let's take a look at the different cases to find the mean:

Case 1: If there are 'n' number of items in a list. The data is {x₁, x₂, x₃, ... x_n}. The Mean is calculated using the formula:

\bold{\bar{x} = \frac{x_1 + x_2 + x_3 +...x_n}{n}}
\bold{\bar{x} = \frac{\sum{x_i}}{{n}}}

Case 2: 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:

\bold{\bar{x} = \frac{f_1x_1 + f_2x_2 + f_3x_3 +...f_nx_n}{f_1+f_2+f_3...f_n}}
\bold{\bar{x} = \frac{\sum{f_ix_i}}{{\sum{f_i}}}}

Case 3: When items of the set are given in the form of a range, for example, 1-10, 10-20, etc. To find the mean, first we need to calculate the class mark for each class interval, and then the mean is calculated using the given formula:

\bold{\bar{x} = \frac{\sum{f_ix_i}}{{\sum{f_i}}}}

Mean of Ungrouped Data

The mean of ungrouped data is the sum of all the observations divided by the total number of observations. Ungrouped data is known as raw data, where the dataset simply contains all the data in no particular order. The following are the steps that are to be followed to find the mean of ungrouped data:

Note down the entire dataset for which the mean is to be calculated.
Now, apply any of the two formulas added below based on the observation of the data.

Mean Formula For Ungrouped Data

The mean formula for ungrouped data is added below,

\bar{x} = \frac{x_1 + x_2 + x_3 +...x_n}{n}
\bar{x} = \frac{f_1x_1 + f_2x_2 + f_3x_3 +...f_nx_n}{f_1+f_2+f_3...f_n}

The mean formula for ungrouped data added above is used to find the mean of ungrouped data, for example,

Example: Calculate the mean for the following set of data: 2, 6, 7, 9, 15, 11, 13, 12.

Solution:

Given,
Observed values 2, 6, 7, 9, 15, 11, 13, 12
Total number of observed values = 8
Using Mean Formula for Grouped Data
Mean = (Sum of observed values in data) / (Total number of observed values in data)
Sum of observed values = 2 + 6 + 7 + 9 + 15 + 11 + 13 + 12 = 75
Total number of observed values = 8
Mean = 75/8
⇒ Mean = 9.375
Therefore, mean for the given observed values = 9.375

Types of Mean

In statistics, there are four types of mean, and they are weighted mean, Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). When not specified, the mean is generally referred to as the arithmetic mean. Let's take a look at all the types of mean:

Arithmetic Mean

The arithmetic mean is calculated for a given set of data by calculating the ratio of the sum of all observed values to the total number of observed values. When the specification of the mean is not given, it is presumed that the mean is an arithmetic mean. The general formula for the arithmetic mean is given as:

Arithmetic Mean = (Sum of observed values)÷(Number of observed values in data)
\bold{\bar{x} = \frac{\sum{f_i}}{{N}}}

Where,

\bar{x} = Arithmetic mean
F_i= Frequency of each data point
N = Number of frequencies.

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:

Solution: (4 + 36 + 45 + 50 + 75)/5 = 210/5 = 42.

Geometric Mean

The geometric mean is calculated for a set of n values by calculating the nth root of the product of all n observed values. It is defined as the nth root of the product of n numbers in the dataset. The formula for the geometric mean is given as:

Geometric Mean = n^th root of (x₁× x₂× x₃× x₄.... n values)
\bold{G.M. = \sqrt[n]{x_1\times x_2\times x_3\times \ldots \times x_n}}

For example: Find the geometric mean of the numbers: 4, 16, 64

Solution:

\bold{G.M. = \sqrt[n]{x_1\times x_2\times x_3\times \ldots \times x_n}}
G.M. = ∛4 × 16 × 64
G.M. = ∛4096
G.M. = ∛4096
G.M. = 16

Harmonic Mean

The harmonic mean is calculated by dividing the number of values in the observed set by the sum of reciprocals of each observed data value. Therefore, the harmonic mean can also be called the reciprocal of the arithmetic mean. The formula for harmonic mean is given as:

Harmonic Mean = (Number of Observed Values) / (1/n₁ + 1/n₂ + 1/n₃+ . . .)
\bold{H.M. = \frac{1}{\frac{\sum{f_i}}{{N}}} = \frac{N}{\sum{f_i}}}

For Example: Find the harmonic mean of the numbers: 4, 5, and 10

Solution:
Harmonic Mean = (Number of Observed Values) / (1/n₁ + 1/n₂ + 1/n₃+ . . .)
Harmonic Mean = 3/ (1/4 + 1/5 + 1/10)
Harmonic Mean = 3/ 0.55
Harmonic Mean = 5.454

Weighted Mean

The Weighted Mean is calculated in certain cases of the dataset when the given set of data has some values more important than others. In the dataset, a weight 'w_i' is connected to each data 'x_i', and the general formula for weighted mean is given as:

\bold{\text{Weighted Mean} = \frac{\sum{w_ix_i}}{\sum{w_i}}}

Where,

x_iis i^th observation, and
w_iis the Weight of i^th observations.

For example: A student has the following grades in two subjects:

Math: 85 (weight 3)
English: 90 (weight 2)

Calculate the weighted mean of the student's grades.

Solution:

Mean of Grouped Data

Grouped data is the set of data that is obtained by forming individual observations of variables into groups. Grouped data is divided into groups. A frequency distribution table is required for the grouped data, which helps showcase the frequencies of the given data. The mean of grouped data can be obtained using three methods. The methods are:

Direct Method
Assumed Mean Method
Step Deviation Method

Calculating Mean Using Direct Method

The direct method is the simplest method to find the mean of grouped data. The mean of grouped data using the direct method can be calculated using the following steps:

Four columns are created in the table. The columns are Class interval, class marks (x_i), frequencies (f_i), the product of frequencies, and class marks (f_ix_i).
Now, calculate the mean of the grouped data using the formula

Mean Formula For Grouped Data (Using Direct Method)

The mean formula for grouped data using the direct method is added below,

\bold{\text{Mean}(\bar{x}) = \frac{\sum f_ix_i}{\sum f_i}}

Example: Calculate the mean height for the following data using the direct method.

Height (in inches)	60 - 62	62 - 64	64 - 66	66 - 68	68 - 70	70 - 72
Frequency	3	6	9	12	8	2

Solution:

As, \bar{x} = \frac{\sum{f_ix_i}}{{\sum{f_i}}}
Height (in inches)
Frequency(f_i)
Midpoint (x_i)
f_i × x_i
60 - 62
3
61
183
62 - 64
6
63
378
64 - 66
9
65
585
66 - 68
12
67
804
68 - 70
8
69
552
70 - 72
2
71
142

∑f_i = 40

∑f_i x_i= 2644

⇒ Mean = 2644/40 = 66.1
Thus, mean height is 66.1 inches.

Calculating Mean Using Assumed Mean Method

When the calculation of the mean for grouped data using the direct method becomes very tedious, then the mean can be calculated using the assumed mean method. To find the mean using the assumed mean method, the following steps are needed:

Five columns are created in the table, i.e., class interval, class marks (x_i), corresponding deviations (d_i= x_i- A) where A is the central value from class marks as assumed mean, frequencies (f_i), and the product of f_i and d_i.
Now, the mean value can be calculated for the given data using the following formula.

Mean Formula For Grouped Data (Using Assumed Mean Method)

The mean formula for grouped data using the assumed mean method is added below,

the \bold{\text{Mean}(\bar{x}) =A + \frac{\sum f_id_i}{\sum f_i}}

Example: Calculate the mean of the following data using the Assumed Mean Method.

Weight (in kg)	40 - 44	44 - 48	48 - 52	52 - 56	56 - 60	60 - 64
Frequency	2	3	5	7	2	1

Solution:

Let us assume the value of mean be A = 53,
and the required table for the given data is as follows for A = 53:
Weight (in kg)
Frequency(f_i)
Midpoint (x_i)
Deviation (d_i = x_i - A)
40 - 44
2
42
-11
44 - 48
3
46
-7
48 - 52
5
50
-3
52 - 56
7
54
1
56 - 60
2
58
5
60 - 64
1
62
9
Add one more column to the table which give product of f_iand d_i :
Weight (in kg)
Frequency(f_i)
Midpoint (x_i)
Deviation (d_i = x_i - A)
f_i × d_i
40 - 44
2
42
-11
-22
44 - 48
3
46
-7
-21
48 - 52
5
50
-3
-15
52 - 56
7
54
1
7
56 - 60
2
58
5
10
60 - 64
1
62
9
9

∑f_i = 20

∑f_i d_i= -32
Thus, Mean = 53 + (-32)/20 = 53 - 1.6 = 51.4
Thus, mean weight of the given data using assumed mean method is 51.4 Kg.

Calculating Mean Using Step Deviation Method

The step deviation method is also famously known as the scale method or the shift of origin method. When finding the mean of grouped data becomes tedious, using step deviation method can be used. The following are the steps that should be followed while using the step deviation method:

Five columns are created in the table. They are class interval, class marks (x_i, here the central value is A), deviations (d_i), u_i = d_i/h (h is class width), and the product of f_i and UI_i.
Now, the mean of the data can be calculated using the following formula

Mean Formula For Grouped Data (Using Step Deviation Method)

The mean formula for grouped data using the step deviation mean method is added below,

\bold{\text{Mean}(\bar{x}) =A + \frac{\sum f_iu_i}{\sum f_i}\cdot h}

Example: Calculate the mean of the following data using the Step Deviation method.

Age(in year)	20-24	24-28	28-32	32-36	36-40	40-44	44-48
Frequency	3	6	8	5	5	2	1

Solution:

Range of the data is 20 to 48, for assumption of mean, lets take average of the range values,
Assumed mean = (20 + 48) /2 = 68/2 = 34
Let's A = 34 be the assumed mean of the data,
Now, using assumed mean value, let's create the table for step deviation as follows:
Age (in years)
Frequency(f_i)
Class Mark(x_i)
Deviation(d_i = x_i - A)
Step Deviation (u_i = d_i/h)
f_i × u_i
20 - 24
3
22
-12
-3
-9
24 - 28
6
26
-8
-2
-12
28 - 32
8
30
-4
-1
-8
32 - 36
5
34
0
0
0
36 - 40
5
38
4
1
5
40 - 44
2
42
8
2
4
44 - 48
1
46
12
3
3
∑f_i = 30 ∑f_i u_i=- 17
Thus, Mean = 34 + 4 × (-17)/30 = 34 + 4 × (0.56) = 34 - 2.26 = 31.74
Thus, mean age of data using step deviation method is 31.74

Arithmetic Mean vs. Geometric Mean

There are key differences between the Arithmetic Mean and Geometric Mean, which can be listed as follows:

Arithmetic Mean	Geometric Mean
The sum of all values divided by the number of values	n^th root of the product of all values
Suitable for symmetrical data with no extreme values	Suitable for data with positive values and extreme values
Sensitive to extreme values	Not sensitive to extreme values
Used for measuring the central tendency of data	Used for measuring the average growth rate
Can be used for both discrete and continuous data	Usually used for continuous data
Additive in nature	Multiplicative in nature
Denoted by "x̄" or "AM"	Denoted by "G" or "GM"

Solved Question on Mean

Question 1: 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 2: 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 3: 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 4: 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.

Marks Obtained	Number of students
100	1
92	3
80	5
75	10
70	1

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
\bar{x} = \frac{f_1x_1 + f_2x_2 + f_3x_3 +...f_nx_n}{f_1+f_2+f_3...f_n}
⇒ 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 5: Calculate the mean of the following dataset.

Height (in inches)	60 - 62	62 - 64	64 - 66	66 - 68	68 - 70	70 - 72	72 - 74	74 - 76
Frequency	2	3	4	6	5	3	1	1

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(f_i)
Class Mark (x_i)
Deviation (d_i)
Step Deviation (u_i)
f_i × u_i
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 ∑f_iu_i = -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.

Practice Questions on Mean

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

Marks	Number of Students
0-10	3
10-20	5
20-30	9
30-40	8
40-50	5

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Article Tags :

Mean in Statistics

Mean

Mean Symbol

Mean Application

Mean Formula

How to Find the Mean?

Mean of Ungrouped Data

Mean Formula For Ungrouped Data

Types of Mean

Arithmetic Mean

Geometric Mean

Harmonic Mean

Weighted Mean

Mean of Grouped Data

Calculating Mean Using Direct Method

Mean Formula For Grouped Data (Using Direct Method)

Calculating Mean Using Assumed Mean Method

Mean Formula For Grouped Data (Using Assumed Mean Method)

Calculating Mean Using Step Deviation Method

Mean Formula For Grouped Data (Using Step Deviation Method)

Arithmetic Mean vs. Geometric Mean

Solved Question on Mean

Practice Questions on Mean

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