Mean for grouped data with Examples

This article will be about How to calculate the mean for grouped data, with formulas and step by step examples.

What is the mean for grouped data

The arithmetic mean for grouped data is an average that is calculated between all the elements of a set. To find the mean in a not grouped data, we just have to sum all the numbers of the set, and then divide the result by the number of elements, but when we work with grouped data (normally grouped in intervals) the steps we have to follow to calculate the mean is different and a little more complicated.

Normally a data could be grouped in order to reduce the number of variables, this is done by grouping the numbers so the variables are not that wide, so we can interpret the data of a more general way. To understand this, we are going to make an example: if somebody compiles ages of all the people he finds on the street, he could have a variable for each year, to say something like: “144 people were 31 years old, 54 were 21yo, 44 were 33yo” and like this with all the ages, but it would be a great number of variables, so what we can do to reduce the number of variables is group the data to say something like this: “345 people were between 30 and 40 years old.” This way we reduce the number of variables and we would have a more general view of the data.

In this examples we are going to use brackets to define the intervals, where the following interval ]20,30] is composed by the numbers between 20 and 30, without taking into account the number 20, because the bracket is opened.

We are going to use tables to represent the data, just like the next table.

Intervals	Frequency
]10 - 20]	34
]20 - 30]	12
]30 - 40]	23

Formula for the mean in grouped data

Mean =
Σ f * x_i/n

In the formula of the mean for grouped data the letter “f” means the frequency of an interval, the x_i variable is the average of the limits of the interval. A limit of an interval is the maximum and minimum number of an interval, for example, in the interval ]20,30] the limits would be 20 and 30. And last the variable “n” is the sum of all the frequencies of all the intervals.

To find the mean in grouped data, the most practical thing is to form a table where we are finding every part of the formula, this way we can use the data found in this table to write the data directly in the formula and solve.

Examples of mean for grouped data

Example 1: In a survey people were asked about how many times did they eat out of their houses last month, the results were:

Meal out of home	Frequency
]0 - 10]	25
]10 - 20]	109
]20 - 30]	43
total	186

We are going to find the mean of these grouped data. To this we are going to find the average value of each interval "x_i".

Average for the first interval
x₁ =
0 + 10/2
x₁ =
10/2
x₁ = 5
Average for the second interval
x₂ =
10 + 20/2
x₂ =
30/2
x₂ = 15
Average for the third interval
x₃ =
20 + 30/2
x₃ =
50/2
x₃ = 25

Intervals	Frequency	x_i
]0 - 10]	34	5
]10 - 20]	109	15
]20 - 30]	43	25
total	186

Now having these data we are going to find the top part of the formula: "f * x_i" for each interval

Interval 1
f * x₁ = 34 * 5
f * x₁ = 170
Interval 2
f * x₂ = 109 * 15
f * x₂ = 1635
Interval 3
f * x₃ = 43 * 25
f * x₃ = 1075

Intervals	Frequency	x_i	f * x_i
]0 - 10]	34	5	170
]10 - 20]	109	15	1635
]20 - 30]	43	25	1075
total	186

And finally we can now find the mean

Mean =
Σ f * x_i/n
Mean =
170 + 1635 + 1075/186
Mean =
2880/186
Mean = 15.48

Example 2: We asked a group of people about how many hours per week they exercise, the results were:

Hours / week	Frequency
]0 - 5]	15
]5 - 10]	63
]10 - 15]	44
]15 - 20]	8
total	130

Having this we can find the x_i values for each interval

Average for the first interval
x₁ =
0 + 5/2
x₁ =
5/2
x₁ = 2.5
Average for the second interval
x₂ =
5 + 10/2
x₂ =
15/2
x₂ = 7.5
Average for the third interval
x₃ =
10 + 15/2
x₃ =
25/2
x₃ = 12.5
Average for the fourth interval
x₄ =
15 + 20/2
x₄ =
35/2
x₄ = 17.5

Hours / week	Frequency	x_i
]0 - 5]	15	2.5
]5 - 10]	63	7.5
]10 - 15]	44	12.5
]15 - 20]	8	17.5
total	130

Now we find the top of the formula: "f * x_i" for every interval

Interval 1
f * x₁ = 15 * 2.5
f * x₁ = 37.5
Interval 2
f * x₂ = 63 * 7.5
f * x₂ = 472.5
Interval 3
f * x₃ = 44 * 12.5
f * x₃ = 550
Interval 4
f * x₄ = 8 * 17.5
f * x₄ = 140

Hours / week	Frequency	x_i	f * x_i
]0 - 5]	15	2.5	37.5
]5 - 10]	63	7.5	472.5
]10 - 15]	44	12.5	550
]15 - 20]	8	17.5	140
total	130

And last we find the mean

Mean =
Σ f * x_i/n
Mean =
37.5 + 472.5 + 550 + 140/130
Mean =
1200/130
Mean = 9.23

Example 3: In a survey that was made in a university in the physics faculty, they asked to a group of students what were their average grades this year (in a scale from 0-10).

Average grade	Frequency
]0 - 2]	5
]2 - 4]	22
]4 - 6]	120
]6 - 8]	211
]8 - 10]	41
total	399

First we find "x_i" for each interval

First interval
x₁ =
0 +2/2
x₁ =
2/2
x₁ = 1
Second interval
x₂ =
2 + 4/2
x₂ =
6/2
x₂ = 3
Third interval
x₃ =
4 + 6/2
x₃ =
10/2
x₃ = 5
Fourth interval
x₄ =
6 + 8/2
x₄ =
14/2
x₄ = 7
Fifth interval
x₄ =
8 + 10/2
x₄ =
18/2
x₄ = 9

Avg. Grades	Frequency	x_i
]0 - 2]	5	1
]2 - 4]	22	3
]4 - 6]	120	5
]6 - 8]	211	7
]8 - 10]	41	9
total	399

Now we calculate the "f * x_i" of the intervals

Interval 1
f * x₁ = 5 * 1
f * x₁ = 5
Interval 2
f * x₂ = 3 * 22
f * x₂ = 66
Interval 3
f * x₃ = 5 * 120
f * x₃ = 600
Interval 4
f * x₄ = 211 * 7
f * x₄ = 1477
Interval 5
f * x₅ = 41 * 9
f * x₅ = 369

Avg. Grades	Frequency	x_i	f * x_i
]0 - 2]	5	1	5
]2 - 4]	22	3	66
]4 - 6]	120	5	600
]6 - 8]	211	7	1477
]8 - 10]	41	9	369
total	399

And then we find the mean of the grouped data

Mean =
? f * x_i/n
Mean =
5 + 66 + 600 + 1477 + 369/399
Mean =
2571/399
Mean = 6.3

Related articles

Mean median and mode Median for grouped data Mode for grouped data Quartiles Standard deviation Frequency table

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