NCERT Solutions Class 10 Maths Chapter 13 Statistics Exercise 13.4

NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.4 will help students in the visual representation of the cumulative frequency distributions of a data set through graphs. These graphs, in the exercise, are also known as the ogives which help in understanding complex data with ease. This topic is important not just to learn statistical concepts but also to conduct higher-level mathematical analysis and solve practical problems. So, let's discuss this all-important subject of Class 10 via the solutions of exercise 13.4.

1.0Download NCERT Solutions Class 10 Maths Chapter 13 Statistics Exercise 13.4 : Free PDF

2.0Introduction to Cumulative Frequency Distribution:

Before moving on to the topic of the graphical representation of cumulative frequency, let us briefly recap the concept of a cumulative frequency distribution. A cumulative frequency distribution is a method of accumulating frequencies from a particular data set in a manner that every frequency represents the cumulative frequency for all the previous classes. It enables us to realise the way the data is building up over a period of time or in various intervals. Visualising such a distribution through graphs helps in the rapid recognition of trends and patterns, and that's why it's a significant area in statistical analysis. 

3.0Exercise Overview 13.4: Key Concepts

Graphical Representation of Cumulative Frequency Distribution: An Overview

Just like cumulative frequency tables, which are of two types — less than and more than type frequency distributions, cumulative frequency is also represented for both of these types. 

The Less Than Type Ogive: 

Exercise 13.4 is mostly focused on the questions related to the plotting of ogives with less than type frequency distribution. In this type, keep these rules in mind while plotting less than type ogive: 

1. Plot the upper limit of the class interval on the x-axis of the graph. 
2. Plot the cumulative frequencies of corresponding frequencies on the y-axis. 
3. Plot the points according to the respective limit and cumulative frequencies and join these points to form a smooth curve. 

Note: The ogive of a less-than-type frequency distribution generally starts from the origin and increases when moved in the right direction. 

The More Than Type Ogive: 

The More Than Type Ogive helps visually represent a dataset which starts with its highest value and moves downwards. The More Than Type ogive differs from the Less Than Type in terms of its x-axis while plotting on the graph. This ogive can be represented by keeping these points in mind: 

1. Plot the lower limit of the class interval on the x-axis of the graph. 
2. Plot the cumulative frequencies of corresponding frequencies on the y-axis. 
3. Plot the points according to the respective limit and cumulative frequencies and join these points to form a smooth curve. 

Note: In more than type frequency distributions, the ogive starts from the highest value of cumulative frequency on the y-axis, which decreases as it moves rightward. 

Ogives and Median of a Dataset: 

The median is an important measure of central tendency in statistics, used to locate the central value of a dataset. The median is generally found with the help of general formulas. However, it can also be calculated with the help of ogives. In a frequency distribution, when both the less than type and more than type ogives are shown, the point of their interactions represents the median of that dataset. The x-axis of this interaction gives the value of the median. 

Alternatively, the Median is also represented when only less than type ogive is shown on the graph. This can be done by representing the $\frac{n}{2}$ (n is the sum of all the frequencies) on the y-axis. After locating this point, drop a perpendicular from this point to intersect on the x-axis. The intersecting point on the x-axis is the median. 

Practice concepts of NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.4 will solidify your grasp and make sure you are perfectly prepared for exams, as well as practical uses of statistics.

4.0NCERT Class 10 Maths Chapter 13 Statistics Exercise 13.4 : Detailed Solutions

1. The following distribution gives the daily income of 50 workers of a factory.

Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.

Sol.

N=50 gives 2N​=25
On the graph, we will plot the points (120,12),(140,26),(160,34),(180,40), (200,50).

2. During the medial check up of 35 students of a class, their weights were recorded as follows:

Draw a less than type ogive for the given data. Hence obtain the median weight from the graph and verify the result by using the formula.

Sol.

To draw the 'less than' type ogive, we plot the points (38,0),(40,3),(42,5),(44,9), (46,14),(48,28),(50,32) and (52,35) on the graph.

Median from the graph =46.5 kg.
median class is (46-48). (See in the table)
We have ℓ=46,f=14,cf=14, N=35 and
h=2.
Median =ℓ+{f2N​−cf​}×h
=46+{14235​−14​}×2=46+21​=46.5 kg
Hence, the median is same as we have noticed from the graph

3. The following table gives production yield per hectare of wheat of 100 farms of a village.

Change the distribution to a more than type distribution and draw its ogive.

Sol.

Now, we will draw the ogive by plotting the points (50,100),(55,98),(60,90), (65,78),(70,54) and (75,16). Join these points by a freehand to get an ogive of 'more than' type.

5.0Benefits of Studying NCERT Solutions Class 10 Maths Chapter 13

• Helps understand Mean, Median, Mode, and data representation.
• Covers syllabus-aligned questions helps to better the exam preparation.
• Helps in competitive exams like NTSE and Olympiads
• This exercise teaches how to draw and analyze ogives thus enhancing the graphical representation skills
• Useful for real-world data analysis in fields like business and research.