Notes-Class-10-Mathematics-1-Chapter-6-Statistic-Maharashtra Board

Topics to be learn : Measures of a central tendency - mean, median and mode from grouped frequency table. Graphical representation of statistical data - histogram, frequency polygon, pie diagram

Introduction of Statistics :

• Statistics is the field of mathematics that collects, presents, analyzes, and interprets numerical data.
• Statistics are utilized in all aspects of life. It has applications in agriculture, botany, biotechnology, bioinformatics, chemistry, physics, economics, commerce, geography, sociology, and medicine, among other fields.
• An experiment can have several outcomes. The record can be used to examine the possibilities for various outcomes. Statistical criteria are used to do this.

Measures of a central tendency :

We usually find a specific property in the numerical data collected in a survey that the scores have a tendency to cluster around a particular score. This score is a representative of the group. The number is called the measure of the central

tendency.

We have studied the measure of the central tendency, namely the mean, median and mode for ungrouped data.

The mean of statistical data = \(\frac{\text{The sum of all scores}}{\text{Total no. of scores}}=\frac{∑_{i=1}^{N}x_i}{N}\)

 (Here, x_i is the i th score)

The mean is denoted by \(\bar{X}\). It represents the average of the given data

\(\bar{X} = \frac{∑_{i=1}^{N}x_i}{N}\)

Mean from classified frequency distribution :

When the number of scores in a data is large, we use some different methods to find the sum. The methods are :

• Direct method
• Assumed mean method
• Step deviation method

(i) Direct method :

Ex. (1) The percentage of marks of 50 students in a test is given in the following table. Find the mean of the percentage

Percentage of marks	0-20	20-40	40-60	60-80	80-100
No. of students	3	7	15	20	5

Solution :

(1) Vertical columns are drawn as shown in the table.

(2) Classes are written in the first column.

(3) The class mark x_i is in the second column.

(4) In the third column, the number of workers, that is frequency (f_i) is written.

(5) In the fourth column, the product (x_i × f_i) for each class is written.

(6) Then  ∑ x_i f_i   is written.

Class (Percentage of marks)	Class mark xi	Frequency (No. of students) fi	Class mark frequency xi fi
0-20	10	3	30
20-40	30	7	210
40-60	50	15	750
60-80	70	20	1400
80-100	90	5	450
Total		N=∑ fi = 50	∑ xi fi = 2840

(7) The mean is found using the formula

Mean = \(\bar{X} = \frac{∑x_if_i}{N}=\frac{2840}{50}\) = 56.8

∴ The mean of the percentage = 56.8

(ii) Assumed mean method :

In the examples solved above, we see that some times the product x_i f_i is large. If the product x_i f_i is large, it becomes difficult to calculate the mean by direct method. Finding the mean by 'assumed mean method' becomes simpler if we use addition and division in this method.

Ex. : The daily sale of 100 vegetable vendors is given in the following table. Find the mean of the sale by assumed mean method.

Daily sale (Rupees)	1000-1500	1500-2000	2000-2500	2500-3000
No. of vendors	15	20	35	30

Steps to calculate mean by this method :

(1) Assumed mean, A. In this example A = 2250. (Generally, the class mark of the

class having maximum frequency is chosen as the assumed mean.)

(2) In column 1, write the class intervals. (Here daily sale)

(3) Class marks are written in the second column.

(4) Values of di = x_i — A, write in the third column.

(5) In column 4, write the frequencies (f_i). Write ∑ f_i at the bottom of this column.

(6) In the fifth column, the product (f_i × d_i) and their sum is written as ∑ f_i d_i.

\(\bar{d}\) and \(\bar{X}\) are calculated using the formulae.

\(\bar{d} = \frac{∑f_id_i}{f_i}\), Mean \(\bar{X}\) = A + \(\bar{d}\)

Class daily sale (Rupees)	Class mark xi	di = xi- A = xi- 2250	Frequency (No. of vendors) fi	Frequency × Deviation fi × di
1000-1500	1250	-1000	15	-15000
1500-2000	1750	-500	20	-10000
2000-2500	2250→A	0	35	0
2500-3000	2750	500	30	15000
Total			N=∑ fi = 100	∑ fi di = -10000

Assumed mean = A = 2250, d_i = x_i - A is the deviation

\(\bar{d} = \frac{∑f_id_i}{f_i}=\frac{10000}{100}\) = —100

∴ Mean  = A + \(\bar{d}\) = 2250 — 100 =2150

The mean of sale is Rs. 2150.

(iii) Step Deviation Method :

This method reduces the calculations still further.

Steps to calculate mean by this method :

(1) In column 1, write the given class interval. (Amount invested in below example).

(2) In column 2, write the corresponding class marks (x_i).

(3) In column 3, write the values of di where d_i = x_i — A.

(4) Find g, the GCD of all d_i. In column 4, write the values of u_i, where

u_i = \(\frac{x_i-A}{g}=\frac{di}{g}\)

(5) In column 5, write the frequencies (f_i) of the given class intervals. Find their sum (∑f_i).

(6) In column 6, write f_i u_i . Find their sum ( ∑f_iu_i).

Find  using the formula, \(\bar{u}=\frac{∑f_iu_i}{f_i}\)

Find mean, \(\bar{X}\) = A + g\(\bar{u}\)

Formula for finding the median of a grouped frequency distribution :

Median = L + \([\frac{\frac{N}{2}-cf}{f}]\) × h

where, L = lower class limit of the median class.

N =sum of frequencies

cf = cumulative frequency of the class preceding the median class.

f =frequency of the median class.

h =class interval of the median class.

Following are the steps in the calculation of median for grouped frequency distribution:

• Prepare a table containing three columns
• Write the given class intervals in column 1 and their respective frequencies in column 2.
• Write cumulative frequency less than type in column 3.
• Find total frequency N = ∑f_i. Calculate \(\frac{N}{2}\)
• Find cumulative frequency less than type which is just greater than or equal to g.
• The corresponding class is median class.
• Calculate the value of the median by using the formula given above.

[Note : If the class intervals are of inclusive type, make them exclusive type by finding class boundaries. ]

Remember :

• If the given classes are not continuous, we have to make them continuous to find out the median.
• It is difficult to write the scores in the asscending order when the number of scores is large. So the data is classified into groups. It is not possible to find the exact median of a classified data, but the approximate median is found by the formula.

Median = L + \([\frac{\frac{N}{2}-cf}{f}]\)× h

Mode for grouped frequency distribution :

Mode : The score repeating maximum number of times in a data is called the mode of the data.

The following formula is used to estimate the mode of grouped data.

Mode = L + \([\frac{f_1-f_0}{2f_1-f_0-f_2}]\)× h

In the above formula,

L = Lower class limit of the modal class.

f₁ = Frequency of the modal class.

f₀ =Frequency of the class preceding the modal class.

f₂ = Frequency of the class succeeding the modal class.

h = Class interval of the modal class.

Remember :

• We have studied the central tendencies mean, median and mode. Before selecting any of these measures, we have to know the purpose of its selection clearly.
• Suppose, we have to judge the performance of five divisions of standard 10 in the internal examination. For the purpose, we have to find the 'mean' of marks of students in each division.
• If we have to make two groups of students in a division based on their marks in the examination, we have to find the 'median' of their marks.
• If a 'bachat' group producing chalks wants to know about the colour of chalks having maximum demand, it will have to choose the 'mode'.

Pictorial representation of statistical data :

The mean, median or mode of a numerical data or analysis of the data is useful to draw some useful inferences. The tabulation is one of the methods representing numerical data in brief. A table does not immediately convey some characteristics of the data. A common man is interested in the important aspects of a data.

Presentation of data :

Pictorial and graphical presentation are attractive methods of data interpretation.

The tree chart below shows different methods of data interpretation.

We have studied some of these methods and graphs in previous standards. Now we will learn a histogram, a frequency polygon and a pie diagram.

(i) Histogram :

Method of drawing a histogram :

• If the given classes are not continuous, make them continuous. Such classes are called extended class intervals.
• Show the classes on the X-axis with a proper scale.
• Show the frequencies on the Y-axis with a proper scale.
• Taking each class as the base, draw rectangles with heights proportional to the frequencies.
• On the X-axis, a mark ‘ ‘ called the krink mark is shown between the origin and the first class.
• It means, there are no observations up to the first class. The mark can be used on the Y-axis also, if needed. This enables us to draw a graph of optimum size.

(ii) Frequency polygon :

A frequency polygon is another way of presentation. Two methods of drawing a frequency polygon is: (a) With the help of a histogram (b) Without the help of a histogram.

(a) With the help of a histogram :

Steps :

• Draw a histogram from the given information.
• Mark the midpoint of the upper side of each rectangle of the histogram.
• Take one additional rectangle of height zero preceding the first rectangle and one more additional rectangle of height zero succeeding the last rectangle. Mark the midpoints of these two rectangles. Such marks will be on the X-axis.
• Join the successive midpoints by line segments and get a closed figure. It is a frequency polygon.

Example : Here, we have used the histogram figure of above example to learn the method of drawing a frequency polygon.

(b) Without the help of a histogram :

Decide the coordinates of each point with respect to each class. Plot the points on a graph paper. Join successive points by a straight line and get a frequency polygon.

(iii) Pie diagram :

In a pie diagram, the numerical data is shown in a circle. Different components of the data are shown by proportional sectors of the circle.

Reading of Pie diagram :

The following example illustrates how a pie chart gives information at a glance.

Figure shows the annual financial planning of a school. From the pie diagram we see that

• 45% of the amount is reserved for educational equipment.
• 35% of the amount is shown for games.
• 10% of the amount is kept for sanitation.
• 10% of the amount is reserved for environment.

To draw a Pie diagram :

• To draw a pie diagram, the whole circle is divided into sectors proportional to the components of the data.
• The measure of central angle of each sector is found by the following formula.
• The measure of central angle of sector θ = \(\frac{\text{Number of scores in component}}{\text{Total number of scores}}\)x 360°
• A circle of a suitable radius is drawn. Then it is divided into sectors such that, the number of sectors is equal to the number of components in the data.