CBSE Class 9 Maths Revision Notes Chapter 14

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.

Class 9 Mathematics Revision Notes for Statistics of Chapter 14

Statistics

1. 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.
1. 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.

1. 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.
1. 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.
1. 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 11-20 II 2 21-30 IIII I 6 31-40 IIII IIII I 11 41-50 IIII IIII IIII III 18 51-60 IIII IIII IIII 14 61-70 IIII IIII 9 71-80 IIII 4 Total 64
1. Some Terms Used in Statistics
2. Raw Numerical Data

The primary information that has been collected is called raw numerical data.

1. 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 73-14 = 59.

1. 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.

1. Class Interval

The range of the class is its class interval.

1. 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.

1. 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) 11-20 2 2 21-30 6 2+6=8 31-40 11 2+6+11=19 41-50 18 2+6+11+18=37 51-60 14 2+6+11+18+14=51 61-70 9 2+6+11+18+14+9=60 71-80 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) 11-20 2 62+2=64 21-30 6 56+6=62 31-40 11 45+11=56 41-50 18 27+18=45 51-60 14 13+14=27 61-70 9 4+9=13 71-80 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.

1. Representation of Statistical Data

The two forms of numerical data are :

1. Diagrammatic representation
2. Graphical representation
3. Diagrammatic Representation

Diagrammatic representation can be done by using :

1. Pie Diagrams
2. Bar Diagrams
1. Graphical representation

Data can be represented in the form of :

1. Histogram
2. Frequency Polygon
3. Ogive Curve
4. Graphical Representation of Statistical Data
5. 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 x-axis, frequencies on the y-axis and rectangles are joined to the class limit and heights are proportional to the frequencies.
1. 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, one-class 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.

1. Recap

Statistical data can be represented in the following ways :

1. Histogram

On the x-axis, class intervals are plotted and cumulative frequencies are on the y-axis. The corresponding rectangles are drawn representing the data.

1. 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.

1. Arithmetic Mean (AM)
2. 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.

1. 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

1. Direct Method for Arithmetic Mean of Grouped Data

Find the arithmetic mean for the given frequency distribution:

 Marks Frequency (fi) Mid Point (xi) fixi 5-15 3 10 30 15-25 4 20 80 25-35 5 30 150 35-45 7 40 280 45-55 9 50 450 55-65 7 60 420 65-75 6 70 420 75-85 5 80 400 85-95 4 90 360 n=xi=50 fixi = 2590

Arithmetic Mean=x=fixin = 259050=51.80

1. For Calculating the Mean Assumed Mean or Short-Cut Method
2. Short-Cut 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.

1. 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.

1. Step-Deviation 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

1. 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.