Chapter 5
Statistics
The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:
Introduction
The initial step in statistics is the collection of data. If a health camp has been organised at your school, each student’s height and weight in your class are measured. In this condition, the investigators collected this data with a definite objective. This is called primary data. You might have seen your class teacher keep an attendance register and mark your attendance at school. In this case, the data gathered by the teacher was already available somewhere. Hence, this data is called secondary data.
After collecting the data, the next crucial step is to understand how data can be presented easily and meaningfully. The accurate presentation of data is important.
E.g., your teacher is recording the marks of 10 students in your math subject.
55 36 95 73 60 42 25 78 75 62
This is called raw data. The students can understand this data, but it will be time-consuming. To simplify, it can be arranged in increasing order.
25 36 42 55 60 62 73 75 78 95
You can now see that the minimum score was 25 and the maximum score was 95, while the range of marks was 95-25 = 70. Frequency means the number of times a value repeats. For example, here, the frequency of data is 1 as they appear only once.
The graphical representation of data is the best way to represent and analyse data collected from any source. There are three types of graphical representation:
- Bar graph
Suppose you were given a graph representing the number of students in your class, their birthday months, and you were asked questions based on that.
Source: Class 9 NCERT Maths
If this data was collected from every student in the class and you were asked to find the month in which maximum students were born, would you be able to find that quickly? Yes, the answer will be August since the bar is the highest. If you have to find the number of students born in January, you need to find the value of the y-axis, i.e., 3 students.
- Histogram
This is another form of graphical representation of data with continuous class intervals. For example, the weight of students in your class was measured, and the data was represented in class intervals of 0.5 kg.Weights (in kg) Number of students 30.5 – 35.5 9 35.5 – 40.5 6 40.5 – 45.5 15 45.5 – 50.5 3 50.5 – 55.5 1 55.5 – 60.5 2 Total 36 This data can be represented as a histogram.
Source: Class 9 NCERT Maths - Frequency polygon
To make a frequency polygon, the mid-points of the class intervals are required, called class-marks.Class mark = Lower limit + Upper limit/2For the table given above, you can also make the frequency polygon.Source: Class 9 NCERT Maths
What will you do if you are given ungrouped data and asked to find certain values? Mean, median, and mode of ungrouped data help in the analysis of such data.
Mean
It is found by adding all the values of the observations and dividing it by the total number of observations. It is denoted by:
Source: Class 9 NCERT Maths
For ungrouped frequency, the mean is calculated by:
Source: Class 9 NCERT Maths
For example, in your class, 30 students take a maths test out of 100, and their scores are given below. How will you find the mean score of your class?
10 20 36 92 95 40 50 56 60 70
92 88 80 70 72 70 36 40 36 40
92 40 50 50 56 60 70 60 60 88
Marks | Number of students (i.e., the frequency) |
10 | 1 |
20 | 1 |
36 | 3 |
40 | 4 |
50 | 3 |
56 | 2 |
60 | 4 |
70 | 4 |
72 | 1 |
80 | 1 |
88 | 2 |
92 | 3 |
95 | 1 |
Total | 30 |
= 10 + 20 + 36 + 92 + 95 + 40 + 50 + 56 + 60 + 70 + 92 + 88 80 + 70 + 72 + 70 + 36 + 40 + 36 + 40 + 92 + 40 + 50 + 50 56 + 60 + 70 + 60 + 60 + 88 = 1779
Mean = 1779/30 = 59.3
Median
If you are given many values and asked to find the middle-most value, what will you do? You must use the concept of median. In an observation, a median means that the given data is divided into exactly two parts. Here, the data given is:
2 3 4 5 0 1 3 3 4 3
First, arrange the data in increasing order, i.e.:
0 1 2 3 3 3 3 4 4 5
Since, the total number of figures given are even (10), median = mean of the values of the n/2 value and n+1/2 value
Here, n = 10
So mean of 5th and 6th value, i.e. 3+3/2 = 3
Median = 3
What if you were given different values, like:
4 1 8 3 0 2 4 6 7
Arrange in increasing order:
0 1 2 3 4 4 6 7 8
Median for even number = Value of n+1/2 observation
Here, n=9, i.e 5th value
Since, 5th value is 4, so median = 4
Mode
What if you were asked to find the mode of the marks (out of 10) obtained by 20 students in the ungrouped data given below? The mode is the most frequently occurring observation. Here is the data:
4, 6, 5, 9, 3, 2, 7, 7, 6, 5, 4, 9, 10, 10, 3, 4, 7, 6, 9, 9.111
First, arrange the numbers in increasing order:
2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 9, 9, 9, 9, 10, 10
9 marks were scored by 4 students, so its frequency is the highest. Hence, the mode is 9.
Q1. Is mean similar to average?
A1. Yes, the mean is found by taking the average of all the known quantities.
Q2. Can mean, median, and mode be the same?
A2. Yes, it is possible. Mean, median, and mode can be the same.
Q3. What is the formula of the mean?
A3. Mean = Sum of all values/ number of all values
Q4. What is frequency?
A4. Frequency refers to the number of times a data has been repeated.