NCERT Solutions Class 10 Maths Chapter 13 Statistics Exercise 13.1
NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.1 helps students understand an important aspect of statistics, the mean of certain data. This exercise will specifically focus on different calculation methods of the mean of grouped data and the representation of this data into specialised tables. The topic is important for understanding basic statistical problems and forming a solid base for more complex real-world scenarios. So, let’s get a deeper insight into this crucial topic of statistics through the solutions of exercise 13.1.
1.0Download NCERT Solutions Class 10 Maths Chapter 13 Statistics Exercise 13.1 : Free PDF
2.0Introduction to the Mean
The mean, or average, is one of the most commonly used measures of central tendency in statistics. Central tendency refers to a value that represents the center of a dataset and is close to most values in the data. Mean is the typical or central value of a dataset. In simple words, the mean provides us with an idea of the "centre" of the data by taking into account all the values in the dataset and computing a balanced average out of this dataset.
Types of Dataset:
This dataset can be divided into two types, which include:
- Ungrouped Data: Ungrouped data is raw data that is not grouped into any intervals or groups. It is a set of individual values or observations. Every data point is enumerated separately, and no class intervals are utilised to aggregate the data.
- Grouped Data: The exercise is mainly based on the mean of this type of data. Grouped data is data grouped or arranged into intervals or classes known as class intervals. The data is classified, and the frequency or number of occurrences for each class is recorded.
3.0Class 10 Maths Chapter 13 Statistics : Key Concepts
Exercise 13.1 involves various formulas for calculating the mean of grouped or ungrouped data. Let’s explore these formulas to easily solve questions in the exercise:
Mean for Ungrouped Data:
For ungrouped data, the mean is obtained by adding up all the individual values and then dividing the total sum by the number of values. The formula for finding the mean in ungrouped data is:
xˉ=n∑xi
Here,
- ∑xi is the sum of all observations of a dataset.
- n is the total number of observations in the set.
Mean for Grouped Data:
For grouped data, the process is a little different and more complex than the ungrouped data. Since the data is classified into class intervals, the mean is found by determining the class marks (the middle of the class intervals) using different methods, including:
Direct Method:
This is the easiest and most used method for finding the mean of grouped data. In this, we simply multiply each class mark(x1) by its corresponding frequency(fi) and then sum up these products(f1xi). Divide this sum by the sum of all the frequencies to get the final answer. Mathematically, we can express this with the following formula:
xˉ=∑f1∑fixi
Here, xi is the class mark of a certain class interval of the data set. This can be calculated as:
xi=2Upper Limit+Lower Limit
This method, though easy, can sometimes result in large calculations; this is why its alternative methods are used for such large datasets.
Assumed Mean Method:
This method is used when the class mark and frequencies are large, and the chances of the calculation getting complicated are high. The method simply includes assuming a mean denoted by a from the class mark, calculating the deviation di for each class mark, and then finally putting these values in the following formula:
xˉ=a+∑f1∑fidi
Here,
- a is the assumed mean.
- di is the deviation for each class mark, calculated as di = xi – a.
Step-Deviation Method:
To further shorten the calculation of the mean, we use the step-deviation method. The method is helpful when the deviation di has a common factor. The method starts by dividing each deviation by class height or width, calculating the step deviation and then putting all these values in the below-mentioned formula to get the final answer:
xˉ=a+h×∑f1∑fiui
Here,
- ui is the step deviation, such that ui=hdi
- h is the height of the class interval.
To understand better and work efficiently on the concepts mentioned above, go through the NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.1 for a thorough list of questions and solutions.
4.0NCERT Class 10 Maths Chapter 13 Statistics Exercise 13.1 : Detailed Solutions
1. A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
Which method did you use for finding the mean, and why?
Sol.
We have, N=Σfi=20 and Σfi=162.
Then mean of the data is
x= N1×ΣfiXi=201×162=8.1
Hence, the required mean of the data is 8.1 plants.
We find the mean of the data by direct method because the figures are small.
2. Consider the following distribution of daily wages of 50 workers of a factory.
Find the mean daily wages of the workers of the factory by using an appropriate method.
Sol.
We have ∑fi=50 and ∑fi=27260
Mean =∑fi∑fixi=5027260
=545.2
3. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs. 18. Find the missing frequency f.
Sol.
We may prepare the table as given below :
It is given that mean =18.
From the table, we have
a=18, N=44+f and ∑fidi=2f−40
Now, mean =a+ N1×Σfidi
Then substituting the values as given above, we have
18=18+44+f1×(2f−40)
⇒0=44+f2f−40⇒f=20.
4. Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.
Sol.
Mean =∑fi∑fixi=302277=75.9.
5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained a varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
Sol.
a=57, h=3, N=400 and Σfiui=25.
By step deviation method,
Mean =a+h× N1×Σfiui=57+3×4001×25 =57.19
6. The table below shows the daily expenditure on food of 25 households in a locality.
Find the mean daily expenditure on food by a suitable method.
Sol.
Mean =∑fi∑fii=255275=211
8. To find out the concentration of SO2 in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below :
Find the mean concentration of SO2 in the air.
Sol.
Mean =∑fi∑fixi=302.96=0.0986
9. A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
Sol.
Mean =∑fi∑fixi=40499=12.475
10. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
Sol.
Mean =∑fi∑fixi=352430=69.43
5.0Benefits of Studying NCERT Class 10 Maths Chapter 13 Exercise 13.1
- Improves analytical thinking by teaching students how to compute measures like mean, median, and mode.
- Helps students learn how to organize raw data into grouped and ungrouped frequency distributions.
- Encourages logical reasoning while solving different types of data-based questions.
- Strengthens the ability to work with data interpretation and probability-based questions.
