In simple words, statistics implies the process of gathering, sorting, examine, interpret and then present the data in an understandable manner so as to enable one to form an opinion of it and take necessary action, if necessary. Examples:
- A teacher collecting students’ marks, organizing them in ascending or descending manner, and calculating the average class marks, or finding the number of students who failed, informing them so that they start working hard.
- Government officials collecting data for the census, and comparing it with previous records to see whether population growth is in control or not.
- Analyzing the number of followers of a particular religion of a country.
Statistical Tools
The most popular tools of statistics are as follows:
- Arithmetic Mean: Also known as average, the arithmetic mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations.
- Median: Such a value as separates the higher and lower values of a given set of statistical data is called the median.
- Mode: Such a value as occurs most frequently in a given series of statistical data is called the mode.
- Standard Deviation: Such a value as indicates the extent to which certain values of a statistical series tend to vary or disperse from its mean or median is called standard deviation.
- Range: Such a value depicts the difference between the highest and lowest values in a series.
- Correlation: Such a statistical tool as helps study the relationship between two variables is called correlation.
Grouped Data
Such data as is expressed in the form of class intervals and not as an individual unit is termed as grouped data. As the name suggests, the observations are grouped together to form intervals, which then are assigned frequencies pertaining to the number of times all the units belonging to that particular interval appear in the given data set. Such intervals make it very easy to analyze the data set on hand and help interpret and communicate effectively and quickly.
Example:
A teacher assigned with the task of marking 60 students’ papers (out of 100 marks) can divide the data set in 10 groups, like students who have scored between 0 and 10 would be put under 0- 10 class interval, those who got between 10 and 20 would be put in 10- 20 interval, and so on until the last group (interval) becomes 90- 100. Such division is shown as follows:
Marks Scored Number of Students 0 – 10 5 10 – 20 10 20 -30 3 30 – 40 10 40 – 50 4 50 – 60 7 60 – 70 9 70 – 80 6 80 – 90 4 90 – 100 2 Alternatively, the teacher could have made 5 class intervals by choosing aa class size of 20, which is shown as follows:
Marks Scored Number of Students 0 – 20 15 20 – 40 13 40 – 60 11 60 – 80 15 80 – 100 6
This method of grouping data makes it so much easier to calculate the measures of central tendency, especially when the data set is large, like in the above case. It would be a tedious process to write down the marks scored by all 60 students, arranging them in ascending order to calculate median, or list them out, add them up and divide it all by 60 to calculate the arithmetic mean. In other words, grouping makes the data set shrink a bit so that the calculation process can be simplified, and results are calculated effectively since there is a chance of error in the case of handling such a large ungrouped data set.
Arithmetic Mean for Grouped Data
The following steps are required in order to calculate the arithmetic mean for grouped data:
- Calculate the mid-points of the class intervals in the given data set. The mid-points or class marks, denoted by ‘m’ are computed by adding up the lower and upper-class limits and dividing the said sum by 2. In other words, one needs to find the average of the upper and lower class limits of a particular class to get the mid-points.
Example:
Class Intervals Class Marks/ Mid- points 0 – 10 = 05 10 – 20 = 15 20 – 30 = 25 30 – 40 = 35 40 – 50 = 45
- Multiply the frequencies of the given class intervals, denoted by ‘f’ with their respective class marks, denoted by ‘m’. After multiplying all the f with respective m, add up all these results to depict it as Σfm.
- Add up all the frequencies together, and denote it with Σf.
- Divide the sum of frequencies (Σf) with the sum of the product of mid-points and frequencies (Σfm).
- The number so obtained is the arithmetic mean for the given data set.
Hence, arithmetic mean for a given data set where class marks are m, and frequencies are f, through direct method is calculated using the following formula:
Sample Questions
Question 1. Calculate the arithmetic mean for the following data set using direct method:
| Marks | Number of Students |
| 0 – 10 |
5 |
| 10 – 20 |
12 |
| 20 – 30 |
14 |
| 30 – 40 |
10 |
| 40 – 50 |
9 |
Solution:
For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:
Marks Number of Students(f) Mid- Points(m) fm
0 – 10 5
5
25
10 – 20 12
15
180
20 – 30 14
25
350
30 – 40 10
35
350
40 – 50 9
45
405
Σf = 50
Σfm = 1310
Mean = X̄ =
= = 26.2 Hence, the mean of the given data set is 26.2
Question 2. Calculate the arithmetic mean for the following data set using the direct method:
| Class Intervals | Frequency |
|
0 – 2 |
2 |
|
2 – 4 |
4 |
|
4 – 6 |
6 |
|
6 – 8 |
8 |
|
8 – 10 |
10 |
Solution:
For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:
Class Intervals Frequency(f) Mid- Points(m) fm
0 – 2
2
1
2
2 – 4
4
3
12
4 – 6
6
5
30
6 – 8
8
7
56
8 – 10
10
9
90
Σf = 30
Σfm = 190 Mean = X̄ =
= = 6.33 Hence, the mean of the given data set is 6.33
Question 3. Calculate the arithmetic mean for the following data set using the direct method:
| Class Intervals | Frequency |
|
10 – 20 |
5 |
|
20 – 30 |
3 |
|
30 – 40 |
4 |
|
40 – 50 |
7 |
|
50 – 60 |
2 |
|
60 – 70 |
6 |
|
70 – 80 |
13 |
Solution:
For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:
Class Intervals Frequency(f) Mid- Points(m) fm
10 – 20
5
15
75
20 – 30
3
25
75
30 – 40
4
35
140
40 – 50
7
45
315
50 – 60
2
55
110
60 – 70
6
65
390
70 – 80
13
75
975
Σf = 40
Σfm = 2080 Mean = X̄ =
= = 52 Hence, the mean of the given data set is 52.
Question 4. Calculate the arithmetic mean for the following data set using the direct method:
| Class Intervals | Frequency |
|
100 – 120 |
4 |
|
120 – 140 |
6 |
|
140 – 160 |
10 |
|
160 – 180 |
8 |
|
180 – 200 |
5 |
Solution:
For the computation of mean, we need to calculate the class intervals of the given class intervals. This is done as follows:
Class Intervals Frequency(f) Mid- Points(m) fm
100 – 120
4
110
440
120 – 140
6
130
780
140 – 160
10
150
1500
160 – 180
8
170
1360
180 – 200
5
190
950
Σf = 33
Σfm = 5030 Mean = X̄ =
= = 152.42 Hence, the mean of the given data set is 152.42