Statistics Class 10 Notes CBSE Maths Chapter 13 (Free PDF Download)

Statistics

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In other words, it is a mathematical discipline to collect and summarize data.

Mean of grouped data

Mean is a measure of central tendency and is defined as an average value of given observations.

There are three methods to find mean of grouped data:

1. Direct Method:

\[\overline x \; = \;\dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]

Where \[\overline x \] is mean, \[f\] is frequency and \[x\] is the mid-interval

2. Assumed Mean Method:

\[\overline x \; = \;A\; + \;\dfrac{{\sum\limits_{i = 1}^n {{f_i}{d_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]

Here, \[A\] is called assumed mean and \[d\; = \;x\; - \;A\]

3. Step Deviation method:

\[\overline x \; = \;A\; + \;h\; \times \;\dfrac{{\sum\limits_{i = 1}^n {{f_i}{t_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]

Here, h is called class size and \[t\; = \;\dfrac{d}{h}\]

Mode of Grouped Data:

It is the observation having the highest value of frequency among all observations.

In case of grouped data, firstly modal class is located and then following formula is used to calculate mode:

\[Mode\; = \;l\; + \;\left( {\dfrac{{{f_1}\; - \;{f_0}}}{{2{f_1}\; - \;{f_0}\; - \;{f_2}}}} \right)\; \times \;h\]

\[\;l\]is lower limit of modal class.

\[h\] is upper limit of modal class.

\[{f_1}\] is frequency of the modal class.

\[{f_0}\] is frequency of the class preceding the modal class.

\[f\] is frequency  of the class succeeding the modal class.

Median of Grouped Data:

It is the value of the middlemost observation of the data.

For grouped data, following formula is used to find median of the data but first we find the median class, then apply formula:

\[Median\; = \;l\; + \;\left( {\dfrac{{\dfrac{N}{2}\; - \;c}}{f}} \right)\; \times \;h\]

\[l\] is Lower limit of the median class

\[f\] is Cumulative frequency preceding the median class frequency

\[N\] is Sum of the frequencies

\[h\] is Width of the class interval

Working Rule:

Step 1: Prepare the table of less than the cumulative frequency using the frequency table given in the question. 

Step 2: Find out the cumulative frequency corresponding to the value of \[\dfrac{N}{2}\]. A median class interval is defined as a class interval corresponding to this cumulative frequency.

Step 3: Find the value of lower limit l and frequency f of the median class.

Step 4: Find the width 'h' of the median class interval.

Step 5: Find value of ‘c’, the cumulative frequency of preceding median class 

Step 6: Apply the formula written above.

Graphs in Statistics:

Graphical Representation of Cumulative Frequency Distribution

We find Cumulative frequency by adding the frequencies of the preceding intervals up to that class interval and the frequency of a class interval.

Ogive (Cumulative Frequency Curve)

There are two ways of constructing an Ogive or cumulative frequency curve. 

1. Less than type Ogive: It is represented by a rising curve.

2. More than type Ogive: It is represented by a falling curve.

Ogive is pronounced as O-jive and This curve is ‘S-shaped.

To Plot an Ogive: 

1. We plot the actual limits along the x-axis and the cumulative frequencies along the y-axis. Plot corresponding coordinates of each class interval.

2. Join the plotted points with a rough hand.

3. Connect an Ogive to a point on the X-axis which represents the actual lower limit of the first class. 

Revision Notes Class 10 Maths Chapter 13: Introduction to Statistics

Class 10 Chapter 13 Maths notes are prepared by the subject experience teachers at Vedantu. It includes all statistical concepts like grouped data, ungrouped data, measures of central tendency like mean, median, mode, methods to find mean, median, and mode, and the relationship between them, along with the important facts and formulae that are important to have a proper understanding of the chapter. 

To have a quick overview of the important concepts of the chapter before the exams, download Class 10 Chapter 13 Maths notes free PDF through the link given here.

A Quick Overview of Class 10 Maths Chapter 13

In Class 10 Maths Chapter 13, students will learn the three measures of central tendency that are mean, median, and mode from ungrouped data to that of grouped data. Students will also understand the concepts of cumulative frequency, the cumulative frequency distribution, and how to draw cumulative frequency known as ogives. 

Mean of Grouped Data

The mean of the observation is the sum of all the values given in the observations divided by the total number of observations.

If y₁,y₂, y₃, y₄...yₙ are observation with respective frequencies f1, f2, f3, f4, ….,fn

Then, the mean of the yof the data is given by:

\[\bar{y}\] = \[\frac{f_{1}y_{1}, f_{2}y_{2}, f_{3}y_{3}, f_{4}y_{4},+.....+y_{n}}{f_{1}, f_{2}, f_{3}, f_{4},....f_{n} }\]

Mode of Grouped Data

The mode of the is that value among the observations which occurs most frequently and is also defined as the value of the observation having a maximum frequency.

Median of Grouped Data

Median is defined as the measure of central tendency which obtains the value of the middlemost observation in the data.

If n is odd, then the median is the \[\left ( \frac{n+1}{2} \right )^{th}\]term of the observations.

If n is even, then the median is defined as the average of the \[\left ( \frac{n}{2} \right )^{th}\] and the \[\left ( \frac{n+1}{2} \right )^{th}\] term of the observations.

Cumulative Frequency

The cumulative frequency of the class is the frequency that is received by adding the frequencies of all the classes above the given class.

List of the Topics and Subtopics Covered in Class 12 Maths Chapter 13

13.1:  Introduction

13.2:  Mean of Grouped Data

13.3:  Mode of Grouped Data

13.4:  Median of Grouped Data

13.5:  Graphical Representation of Cumulative Frequency Distribution

To have a detailed understanding of all the above topics discussed above, download Class 10 Maths Statistics Notes free PDF now.

Benefits of Class 10 Maths Chapter Statistics Notes

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• The Class 10 Maths Chapter Statistics notes are 100% accurate as the notes are prepared by the experts after having extensive research of the topics.

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Solved Examples

1. The daily expenditure on food for 25 households in a locality is depicted in the table given below. Determine the mean of daily expenditure on food using an appropriate method.

Solution:

Determine the mid-point of the given interval using the formula.

Midpoint (xi) = (upper limit + lower limit)/2

Let the mean (A) be equal to 225.

Class size (h) = 50

Mean = Σ fi.xi / Σ fi

 = 5275 / 25 = 211

Hence, the mean of daily expenditure on food comes out to be 211.

2. Information on the observed lifetimes (in hours) of 225 electrical components is given in the following data. Calculate the modal lifetimes of the components.

Solution:

Based on the given data, the modal class is 60–80 as the frequency of this interval is maximum..

So, lower limit of modal class l = 60,

The frequencies are as follows:

fm = 61, f1 = 52, f2 = 38 and h = 20

Mode = l+ [(fm-f1)/(2fm-f1-f2)]×h

Substituting the values in the formula, we obtain

Mode =60+[(61-52)/(122-52-38)]×20

Mode = 60+((9 x 20)/32)

Mode = 60+(45/8) = 60+ 5.625

Hence, the modal lifetime of the components is computed to be 65.625 hours.

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Conclusion

Statistics is one of the most important and easy to understand the chapter in the CBSE Class 10 Maths syllabus. Learning statistics will enable students to properly collect and use data, make accurate analyses, and prepare a great presentation of results. Solved examples and revision notes help students achieve desired results in their exams in terms of scoring and we take pride in improving the conceptualisation of topics in students. We advise students to go through these revision notes along with the other related links that have been provided in this article so that they can make full use of the materials provided by Vedantu.

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