Mean, Median and Mode of Grouped Data
Suppose we want to compare the age of students in two schools and determine which school has more aged students. If we compare on the basis of individual students, we cannot conclude anything. However, if for the given data, we get a representative value that signifies the characteristics of the data, the comparison becomes easy.
A certain value representative of the whole data and signifying its characteristics is called an average of the data. Three types of averages are useful for analyzing data. They are:
- Mean
- Median
- Mode
In this article, we will study three types of averages for the analysis of the data.
Table of Content
- Mean
- Methods for Calculating Mean for Grouped Data
- Median
- Median Formula for Grouped Data
- Mode
- Mode Formula for Grouped Data
- Ogives
- Types of Ogives
- Mean, Median, and Mode of Grouped Data β FAQs
Mean
The mean (or average) of observations is the sum of the values of all the observations divided by the total number of observations. The mean of the data is given by x = f1x1 + f2x2 + β¦. + fnxn/f1 + f2+β¦ + fn. The mean Formula is given by,
Mean = β(fi.xi)/βfi
Methods for Calculating Mean for Grouped Data
To calculate mean of grouped data, we can use following methods:
- Direct Method
- Assumed β Mean Method
- Step-Deviation Method
Letβs discuss about these methods in detail as follows:
Method 1: Direct Method for Calculating Mean
To find mean using direct method, we can use the following steps:
Step 1: For each class, find the class mark xi, as
x=1/2(lower limit + upper limit)
Step 2: Calculate fi.xi for each i.
Step 3: Use the formula Mean = β(fi.xi)/βfi.
Example: Find the mean of the following data.
Class Interval |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
---|---|---|---|---|---|
Frequency |
12 |
16 |
6 |
7 |
9 |
Solution:
We may prepare the table given below:
Class Interval
Frequency fi
Class Mark xi
( fi.xi )
0-10
12
5
60
10-20
16
15
240
20-30
6
25
150
30-40
7
35
245
40-50
9
45
405
βfi=50
βfi.xi=1100
Mean = β(fi.xi)/βfi = 1100/50 = 22
Method 2: Assumed β Mean Method For Calculating Mean
For calculating the mean in such cases we proceed as under.
Step 1: For each class interval, calculate the class mark x by using the formula: xi = 1/2 (lower limit + upper limit).
Step 2: Choose a suitable value of mean and denote it by A. x in the middle as the assumed mean and denote it by A.
Step 3: Calculate the deviations di = (x, -A) for each i.
Step 4: Calculate the product (fi x di) for each i.
Step 5: Find n = βfi
Step 6: Calculate the mean, x, by using the formula: X = A + βfidi/n.
Example: Using the assumed-mean method, find the mean of the following data:
Class Interval |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
---|---|---|---|---|---|
Frequency |
7 |
8 |
12 |
13 |
10 |
Solution:
Let A=25 be the assumed mean. Then we have,
Class Interval Frequency
fi
Mid value
xi
Deviation
di=(xi-25)
(fixdi) 0-10 7 5 -20 -140 10-20 8 15 -10 -80 20-30 12 25=A 0 0 30-40 13 35 10 130 40-50 10 45 20 200 βfi = 50 β(fixdi) = 100 Mean = X = A + βfidi/n = (25 + 110/50) = 27.2
Check: Arithmetic Mean
Method 3: Step-Deviation Method for Calculating Mean
When the values of x, and f are large, the calculation of the mean by the above methods becomes tedious. In such cases, we use the step-deviation method, given below.
Step 1: For each class interval, calculate the class mark x, where X = 1/2 (lower limit + upper limit).
Step 2: Choose a suitable value of x, in the middle of the x, column as the assumed mean and denote it by A.
Step 3: Calculate h = [(upper limit) β (lower limit)], which is the same for all the classes.
Step 4: Calculate ui = (xi -A) /h for each class.
Step 5: Calculate fu for each class and hence find β(fi Γ ui).
Step 6: Calculate the mean by using the formula: x = A + {h Γ β(fi Γ ui)/ βfi}
Example: Find the mean of the following frequency distribution:
Class |
50-70 |
70-90 |
90-110 |
110-130 |
130-150 |
150-170 |
---|---|---|---|---|---|---|
Frequency |
18 |
12 |
13 |
27 |
8 |
22 |
Solution:
We prepare the given table below,
Class
Frequency
fi
Mid Value
xi
ui=(xi-A)/h
=(xi-100)20
(fixui)
50-70
18
60
-2
-36
70-90
12
80
-1
-12
90-110
13
100=A
0
0
110-130
27
120
1
27
130-150
8
140
2
16
150-170
22
160
3
66
βfi = 100
β(fixui) = 61
A = 100, h = 20, βfi = 100 and β(fi x ui) = 61
x = A + {h Γ β(fi x ui) / βfi}
=100 + {20 Γ 61/100} = (100 + 12.2) =112.2
Check: What is mean method formula?
Median
We first arrange the given data values of the observations in ascending order. Then, if n is odd, the median is the (n+1/2). And if n is even, then the median will be the average of the n/2th and the (n/2 +1)th observation.
Median Formula for Grouped Data
The formula for Calculating Median:
Median, Me = l + {h x (N/2 β cf )/f}
Where,
- l = lower limit of median class.
- h = width of median class.
- f = frequency of median class,
- cf = cumulative frequency of the class preceding the median class.
- N = βfi
Example: Calculate the median for the following frequency distribution.
Class Interval |
0-8 |
8-16 |
16-24 |
24-32 |
32-40 |
40-48 |
---|---|---|---|---|---|---|
Frequency |
8 |
10 |
16 |
24 |
15 |
7 |
Solution:
We may prepare cumulative frequency table as given below,
Class
Frequency
Cumulative Frequency
0-8
8
8
8-16
10
18
16-24
16
34
24-32
24
58
32-40
15
73
40-48
7
80
N = βfi = 80
Now, N = 80 = (N/2) = 40.
The cumulative frequency just greater than 40 is 58 and the corresponding class is 24-32.
Thus, the median class is 24-32.
l = 24, h = 8, f = 24, cf = c.f. of preceding class = 34, and (N/2) = 40.
Median, Me = l+ h{(N/2-cf)/f}
= 24 + 8 {(40 β 34)/ 24}
= 26
Hence, median = 26.
Check: Central Tendency
Mode
Mode is that value of a variety that occurs most often. More precisely, the mode is the value of the variable at which the concentration of the data is maximum.
Modal Class: In a frequency distribution, the class having the maximum frequency is called the modal class.
Mode Formula for Grouped Data
The formula for Calculating Mode:
Mo = xk + h{(fk β fk-1)/(2fk β fk-1 β fk+1)}
Where,
- xk = lower limit of the modal class interval.
- fk = frequency of the modal class.
- fk-1= frequency of the class preceding the modal class.
- fk+1 = frequency of the class succeeding the modal class.
- h = width of the class interval.
Example 1: Calculate the mode for the following frequency distribution.
Class |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
70-80 |
---|---|---|---|---|---|---|---|---|
Frequency |
5 |
8 |
7 |
12 |
28 |
20 |
10 |
10 |
Solution:
Class 40-50 has the maximum frequency, so it is called the modal class.
xk = 40, h = 10, fk = 28, fk-1 = 12, fk+1 = 20
Mode, Mo= xk + h{(fk β fk-1)/(2fk β fk-1 β fk+1)}
= 40 + 10{(28 β 12)/(2 Γ 28 β 12 β 20)}
= 46.67
Hence, mode = 46.67
Important Result
Relationship among mean, median and mode,
Mode = 3(Median) β 2(Mean)
Example 2: Find the mean, mode, and median for the following data,
Class |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
Total |
---|---|---|---|---|---|---|
Frequency |
8 |
16 |
36 |
34 |
6 |
100 |
Solution:
We have,
Class
Mid Value xi
Frequency fi
Cumulative Frequency
fi . xi
0-10
5
8
8
40
10-20
15
16
24
240
20-30
25
36
60
900
30-40
35
34
94
1190
40-50
45
6
100
270
βfi=100
βfi. xi=2640
Mean = β(fi.xi)/βf
= 2640/100
= 26.4
Here, N = 100 β N / 2 = 50.
Cumulative frequency just greater than 50 is 60 and corresponding class is 20-30.
Thus, the median class is 20-30.
Hence, l = 20, h = 10, f = 36, c = c. f. of preceding class = 24 and N/2=50
Median, Me = l + h{(N/2 β cf)/f}
= 20+10{(50-24)/36}
Median = 27.2.
Mode = 3(median) β 2(mean) = (3 Γ 27.2 β 2 Γ 26.4) = 28.8.
Check: Mean, Median and Mode
Ogives
Let a grouped frequency distribution be given to us. On graph paper, we mark the upper-class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.
By joining these points successively by line segments, we get a polygon, called a cumulative frequency polygon. On joining these points successively by smooth curves, we get a curve, known as ogives.
The step-by-step process for the construction of an Ogive graph:
- First, we mark the upper limits of class intervals on the horizontal x-axis and their corresponding cumulative frequencies on the y-axis.
- Now plot all the corresponding points of the ordered pair given by (upper limit, cumulative frequency).
- Join all the points with a free hand.
- The curve we get is known as ogive.
Types of Ogives
- Less Than Ogive
On graph paper, we mark the upper-class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.
- On joining these points successively by line segments, we get a polygon, called a cumulative frequency polygon.
- On joining these points successively by smooth curves, we get a curve, known as less than ogive.
- More Than Ogive
On graph paper, we mark the lower class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.
- On joining these points successively by line segments, we get a polygon, called the cumulative frequency polygon.
- On joining these points successively by smooth curves, we get a curve, known as more than ogive.
Practice Problems on Mean, Median and Mode of Grouped Data
Problem 1: The following table shows the distribution of marks obtained by students in a test:
Marks | Number of Students |
---|---|
0 β 10 | 5 |
10 β 20 | 8 |
20 β 30 | 12 |
30 β 40 | 7 |
40 β 50 | 3 |
Calculate the mean marks of the students.
Problem 2: A survey was conducted to find the number of hours teenagers spend on social media per week. The data is presented below:
Hours | Frequency |
---|---|
0 β 5 | 4 |
5 β 10 | 6 |
10 β 15 | 10 |
15 β 20 | 12 |
20 β 25 | 8 |
Determine the mean number of hours spent on social media.
Problem 3: The heights of students in a school are grouped as follows:
Height (cm) | Frequency |
---|---|
120 β 130 | 10 |
130 β 140 | 14 |
140 β 150 | 16 |
150 β 160 | 8 |
160 β 170 | 2 |
Calculate the median height of the students.
Problem 4: The following table shows the distribution of monthly incomes of families in a town:
Income (in $) | Frequency |
---|---|
2000 β 3000 | 20 |
3000 β 4000 | 25 |
4000 β 5000 | 15 |
5000 β 6000 | 10 |
6000 β 7000 | 5 |
Determine the median monthly income.
Problem 5: The following table shows the scores of students in a mathematics exam:
Scores | Frequency |
---|---|
0 β 20 | 3 |
20 β 40 | 7 |
40 β 60 | 12 |
60 β 80 | 18 |
80 β 100 | 10 |
Calculate the mode of the scores.
Problem 6: A shopkeeper recorded the sales of different quantities of an item in a week:
Quantity Sold | Frequency |
---|---|
0 β 10 | 5 |
10 β 20 | 9 |
20 β 30 | 15 |
30 β 40 | 10 |
40 β 50 | 6 |
Determine the mode of the quantity sold.
Mean, Median, and Mode of Grouped Data β FAQs
What is Grouped Data?
Grouped data is data that has been organized into groups known as classes. It is often used in statistical analysis to simplify data and focus on key trends and patterns.
How Do You Calculate the Mean of Grouped Data?
To calculate the mean of grouped data, multiply the midpoints of each class by their respective frequencies, sum these products, and then divide by the total number of data points.
Can You Find the Median in Grouped Data?
Yes, the median can be found in grouped data. It is the value that divides the dataset into two equal parts. For grouped data, the median class is identified first, and then the median value is calculated using a formula that considers the cumulative frequency.
How is the Mode Determined for Grouped Data?
The mode for grouped data is the value or class with the highest frequency. For a more precise value within a class, a modal class is identified, and the mode is calculated using a formula that accounts for the frequencies of the modal class and its adjacent classes.
Is It Possible to Calculate All Three Measures for All Types of Data?
Yes and no. Continuous data can have a mean, median, and mode calculated. Ordinal data can have a median and mode, while nominal data typically only has a mode.
Why Are These Measures Important?
Mean, median, and mode are measures of central tendency that provide insights into the distribution, concentration, and spread of data, helping in making informed decisions and comparisons.
How Do You Handle Outliers in These Calculations?
Outliers can skew the mean significantly. While the mean is sensitive to outliers, the median and mode are more robust measures that are less affected by extreme values.
Can These Measures Be Used for Comparative Analysis?
Yes, mean, median, and mode are often used to compare different datasets, such as comparing the average performance of students across different schools.
What Are the Limitations of Using Mean, Median, and Mode?
Each measure has its limitations. The mean is affected by outliers, the median may not reflect the distributionβs shape, and the mode can be ambiguous in datasets with multiple peaks.
How Do You Interpret These Measures in Real-World Data?
Interpreting these measures depends on the context. For example, the mean income of a region may give an idea of economic status, while the median can indicate the income level that divides the population into two equal groups, and the mode can show the most common income level.
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