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Class 9 Mathematics Revision Notes for Statistics of Chapter 14
Mathematics Revision Notes for Class 9 Chapter 14 Statistics of Extramarks are curated by subject matter experts according to the NCERT curriculum. Students can refer to these notes for a conceptual understanding of the topics explained in Class 9 Mathematics Chapter 14.
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Access CBSE Class 9 Mathematics Chapter 14–Statistics Notes
Statistics
 Introduction
 The branch of mathematics concerned with the collection, classification, tabulation, representation, reasoning, testing of data and drawing inferences from it is referred to as statistics.
 Data is often expressed in the form of graphs, tables, etc. Statistical methods are used, like estimation or prediction.
 Statistics are used to organise and process numerical data in a systematic manner.
 The interrelation of statistics with biology, psychology, economics, trade etc. can be used to interpret and analyse the data from different subjects.
 Measures of central tendencies are expressions that give some information about numerous numerical data.
 An average represents the middle value out of a group of values, which has extremities on both sides.
 Numerical Data and Its Representation
The scores of 64 students in a class test out of 100 are as follows :
Table 1
58  38  52  47  16  50  61  37 
44  55  38  49  44  52  67  51 
33  48  23  51  56  61  46  41 
65  43  71  29  50  56  68  25 
55  49  44  73  23  63  41  42 
66  59  52  28  50  56  60  38 
40  73  45  30  47  40 
This data is presented in an organised manner to present the scores of students in the class. This is the tabular form of presenting data; many other methods can be used.
 Arranged Data
Arranged Numerical Data
The data from Table 1 is arranged in ascending order.
Table 2
14  16  23  23  25  28  29  30 
33  37  37  38  38  38  40  40 
40  40  40  41  41  42  43  44 
44  44  45  46  46  47  47  48 
49  49  50  50  50  51  51  52 
52  52  55  55  56  56  56  58 
58  59  60  61  61  62  63  65 
66  67  68  68  71  72  73  73 
Information gathered from the table:
The minimum marks are 14 and the maximum marks are 73. There is a repetition of scores i.e 40 marks, occurring 5 times, making it the maximum repetition.
Drawbacks:
 This method is very difficult to perform.
 Large amounts of data cannot be arranged and put in an ascending order.
 No crucial information will be gained from this statistical method.

Ungrouped Frequency Distribution Table
The data from table 1 is used to create an ungrouped frequency distribution table. Numbers are written from smallest to largest and repeated numbers are marked using tally marks. Tally marks correspond to the frequency of the number occurring in the data.
Table 3
Tally Marks  Frequency  Marks 
14  I  1 
15  
16  I  1 
17  
18  
19  
20  
21  
22  
23  II  2 
24  
25  I  1 
26  
27  
28  I  1 
Tally Marks  Frequency  Marks 
29  I  1 
30  I  1 
31  
32  
33  I  1 
34  
35  
36  
37  II  2 
38  III  3 
39  IIII  5 
40  IIII  5 
41  II  2 
42  I  1 
43  I  1 
Tally Marks  Frequency  Marks 
44  III  3 
45  I  1 
46  II  2 
47  II  2 
48  I  1 
49  II  2 
50  III  3 
51  II  2 
52  III  3 
53  
54  
55  II  2 
56  III  3 
57  
58  I  1 
Tally Marks  Frequency  Marks 
59  I  1 
60  I  1 
61  II  2 
62  I  1 
63  I  1 
64  
65  I  1 
66  I  1 
67  I  1 
68  II  2 
69  
70  
71  I  1 
72  II  2 
73  II  2 
Conclusions can be drawn more easily from such a table, as mere observation can produce statistical expression. It is obvious from the table that there are a lot of students who received grades between 44 and 58.
Drawback
 A grouped frequency table can be used for a smaller span to represent data.
 Grouped Frequency Distribution Table
Distribution of data into organised groups and classes using tally marks is done in a grouped frequency distribution table.
Classes taken are from 11 to 20, 21 – 30…. 71 – 80.
Table 4
Class  Tally Marks  Frequency 
1120  II  2 
2130  IIII I  6 
3140  IIII IIII I  11 
4150  IIII IIII IIII III  18 
5160  IIII IIII IIII  14 
6170  IIII IIII  9 
7180  IIII  4 
Total  64 
 Some Terms Used in Statistics
 Raw Numerical Data
The primary information that has been collected is called raw numerical data.
 Range of the Data
The difference between the largest and smallest value in the data is called its range.
Example: The range of data in table 2 is 7314 = 59.
 Class Limit
The smallest and largest possible data values for each class are represented by class limits. The smallest possible value in a class is its lower limit and the largest possible value in a class is its upper limit.
 Class Interval
The range of the class is its class interval.
 Frequency of the Class
The frequency of a class interval is the number of observations that occur in the interval. Thus, it can be represented as tally marks or as a count.
 Cumulative Frequency Table
The cumulative frequency is the frequency of observations less than a certain class interval’s upper limit.
Table 5
Class  Frequency  Cumulative Frequency
(Less than the upper class limit) 
1120  2  2 
2130  6  2+6=8 
3140  11  2+6+11=19 
4150  18  2+6+11+18=37 
5160  14  2+6+11+18+14=51 
6170  9  2+6+11+18+14+9=60 
7180  4  2+6+11+18+14+9+4=64 
Total= 64 
In this table, the column of cumulative frequency shows the number of scores less than the upper class limit of the particular class. Thus, such a table is called ‘a cumulative frequency less than’ table.
Along similar lines, the cumulative frequency more than the lower limit of a class is equal to the sum of the frequency of that particular class and the frequencies of all the classes succeeding to it. Table 6 given below shows the cumulative frequency of this type.
Table 6
Class  Frequency  Cumulative Frequency
(More than the lower class limit) 
1120  2  62+2=64 
2130  6  56+6=62 
3140  11  45+11=56 
4150  18  27+18=45 
5160  14  13+14=27 
6170  9  4+9=13 
7180  4  4 
Total= 64 
The cumulative frequency column in this table displays the number of scores that are higher than the respective class’s lower limit. Consequently, a table of this type is known as ‘a cumulative frequency more than’ table.
To create a table like this, make a table with the classes and matching frequencies. From the table’s bottom to its top, jot down the cumulative frequencies. The cumulative frequency for the last class, which is 71 to 80, is 4, making that class’s frequency 4. The frequency of the class before is 9, and it ranges from 61 to 70.
Therefore, 4+9=13 represents the class 71 to 80 cumulative frequency.
 Representation of Statistical Data
The two forms of numerical data are :
 Diagrammatic representation
 Graphical representation
 Diagrammatic Representation
Diagrammatic representation can be done by using :
 Pie Diagrams
 Bar Diagrams
 Graphical representation
Data can be represented in the form of :
 Histogram
 Frequency Polygon
 Ogive Curve
 Graphical Representation of Statistical Data
 Histogram
 A bar diagram demonstrating a continuous frequency distribution, in graphical form, is called a histogram.
 This method involves construction from frequency data, indicating a vertical demonstration of frequencies and classes on the horizontal scale.
 Individual intervals are represented by bars.
 Construction of a histogram is done by using grouped frequency distribution tables. Class limits are shown on the xaxis, frequencies on the yaxis and rectangles are joined to the class limit and heights are proportional to the frequencies.
 Frequency Polygon
 Data in a frequency polygon is represented by plotting the class mark on the horizontal axis and the frequency of the class on the vertical axis.
 The two points are then connected and completed by class marks, oneclass width on either end and a frequency of zero on both ends.
 The construction of histograms is crucial for creating a frequency polygon, by joining the middle points of the upper horizontal sides of the rectangles in the histogram.
 Assumptions are taken such that the frequency of each of the classes before the first class and the last class is taken, and is considered zero.
Frequency polygons can be made without using histograms.
 Recap
Statistical data can be represented in the following ways :
 Histogram
On the xaxis, class intervals are plotted and cumulative frequencies are on the yaxis. The corresponding rectangles are drawn representing the data.
 Frequency polygon
After drawing the histogram, connect the midpoints of the rectangles of the histogram with straight line segments. This gives us the frequency polygon.
 Arithmetic Mean (AM)
 Arithmetic Mean for Ungrouped Data
From the given raw data, the value obtained by summing up all the values of a given variable divided by the total number of values is called Arithmetic Mean.
Let ‘n’ be the total number of values and x1, x2, … xn be the recorded values of the variable then the arithmetic mean is given as follows:
Arithmetic Mean (AM) = x = x1+x2+x3+…+xnn
Or x = xin
The symbol denotes that the values of the given variable are summed over all the given values of x.
 Direct Method for Arithmetic Mean of Ungrouped Data
Arithmetic Mean (AM) = x = fixin
Example: Find the AM of the following data:
Marks  Frequency (fi)  fixi 
7  3  21 
19  4  76 
31  5  155 
40  7  280 
49  9  441 
62  7  434 
73  6  438 
83  5  415 
91  4  364 
n=50  i=1i=50fix1= 2624 
Here, AM = x = fixin = 262450 = 52.48
 Direct Method for Arithmetic Mean of Grouped Data
Find the arithmetic mean for the given frequency distribution:
Marks  Frequency (fi)  Mid Point (xi)  fixi 
515  3  10  30 
1525  4  20  80 
2535  5  30  150 
3545  7  40  280 
4555  9  50  450 
5565  7  60  420 
6575  6  70  420 
7585  5  80  400 
8595  4  90  360 
n=xi=50  fixi = 2590 
Arithmetic Mean=x=fixin = 259050=51.80
 For Calculating the Mean Assumed Mean or ShortCut Method
 ShortCut Method for Ungrouped Data
Here, a value that is roughly in the middle is taken and considered as Assumed Mean (A). On having two middle values, the one with a higher frequency is taken. Then the Arithmetic Mean is given as
Arithmetic Mean = x = A + fididi
where A is the assumed mean and d is the deviation of x from the assumed mean A.
 Shortcut Method for Grouped Data
Here too, similar to the method for ungrouped data, the assumed mean is taken from the given data from the mid values of the table and then the arithmetic mean is obtained as follows:
Arithmetic Mean (AM) = x = A + fidifi
where A is the assumed mean and d is the deviation of x from the assumed mean A.
 StepDeviation Method
In this method,
Arithmetic Mean (AM) = x = A + fiuifi h
where A is the assumed mean, h is the class size(upper limit – lower limit), u is x – Ah
 Median and Mode
Median
On arranging the given statistical data in ascending or descending order of their numerical values, the number in the middle is termed the median.
Let n be the number of values,
Then Median = (n2 + 1)th term if n is odd
And Median = (n2)th term + (n2 + 1)th term 2 if n is even
Mode
The value which appears the maximum number of times among the given statistical data has been termed as ‘mode’. Thus, it is the value with the highest frequency.
FAQs (Frequently Asked Questions)
1. How can arithmetic mean be found for a group of data?
The formula for calculating the arithmetic mean is as follows:
Arithmetic Mean (AM) = x = x1+x2+x3+…+xnn
Arithmetic mean can be found for both grouped and ungrouped data.
2. Why refer to Class 9 Chapter 14 Mathematics Notes by Extramarks?
Curated by subject matter experts, the revision notes for Class 9 Mathematics Chapter 14 offer detailed explanations of the concepts related to the statistical representation of data. Points are neatly organised and tailormade to suit the students’ understanding. Examples are included which enhance the learning experience and allow quick doubtsolving for students. They can access these notes easily from the website for better clarification of concepts.