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”).
• For a population mean, the symbol μ (mu) is used.

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.

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.

1. 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:

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

• \bold{\bar{x} = \frac{\sum{x_i}}{{n}}}

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.

Formula (when each value has a frequency)

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}}}}

OR

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

2. 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:

1. Direct Method
2. Assumed Mean Method
3. 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_i x_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.

Solution:

As, \bar{x} = \frac{\sum{f_ix_i}}{{\sum{f_i}}}

Height (in inches)	Frequency(fi)	Midpoint (xi)	fi × xi
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
	∑fi = 40		∑fi xi = 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.

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(fi)	Midpoint (xi)	Deviation (di = xi - 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(fi)	Midpoint (xi)	Deviation (di = xi - A)	fi × di
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
	∑fi = 20			∑fi di = -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.

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(fi)	Class Mark(xi)	Deviation(di = xi - A)	Step Deviation (ui = di/h)	fi × ui
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
	∑fi = 30				∑fi ui =- 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

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.

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:

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_i is i^th observation, and
• w_i is 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)

Solution:

Step 1: Multiply each grade by its weight.

• Math: 85 × 3 = 255
• English: 90 × 2 = 180

Step 2: Add the weighted values.

∑(w_ix_i) = 255 + 180 = 435

Step 3: Add the weights.

∑w_i = 3 + 2 = 5

Step 4: Apply the formula.

Weighted Mean = 435/5 = 87

Arithmetic Mean vs Geometric Mean

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

Solved Question on Mean

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.

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

\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 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 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