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 * xin
In the formula of the mean for grouped data the letter “f” means the frequency of an interval, the xi 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 "xi".
- Average for the first interval
- x1 = 0 + 102
- x1 = 102
- x1 = 5
- Average for the second interval
- x2 = 10 + 202
- x2 = 302
- x2 = 15
- Average for the third interval
- x3 = 20 + 302
- x3 = 502
- x3 = 25
Intervals | Frequency | xi |
---|---|---|
]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 * xi" 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 | xi | f * xi |
---|---|---|---|
]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 * xin
- Mean = 170 + 1635 + 1075186
- Mean = 2880186
- 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 xi values for each interval
- Average for the first interval
- x1 = 0 + 52
- x1 = 52
- x1 = 2.5
- Average for the second interval
- x2 = 5 + 102
- x2 = 152
- x2 = 7.5
- Average for the third interval
- x3 = 10 + 152
- x3 = 252
- x3 = 12.5
- Average for the fourth interval
- x4 = 15 + 202
- x4 = 352
- x4 = 17.5
Hours / week | Frequency | xi |
---|---|---|
]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 * xi" 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 | xi | f * xi |
---|---|---|---|
]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 * xin
- Mean = 37.5 + 472.5 + 550 + 140130
- Mean = 1200130
- 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 "xi" for each interval
- First interval
- x1 = 0 +22
- x1 = 22
- x1 = 1
- Second interval
- x2 = 2 + 42
- x2 = 62
- x2 = 3
- Third interval
- x3 = 4 + 62
- x3 = 102
- x3 = 5
- Fourth interval
- x4 = 6 + 82
- x4 = 142
- x4 = 7
- Fifth interval
- x4 = 8 + 102
- x4 = 182
- x4 = 9
Avg. Grades | Frequency | xi |
---|---|---|
]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 * xi" 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 | xi | f * xi |
---|---|---|---|
]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 * xin
- Mean = 5 + 66 + 600 + 1477 + 369399
- Mean = 2571399
- Mean = 6.3