NCERT Class 10 Maths Solutions: Statistics

Question:

To find the mean of grouped data, we use the formula: Mean = Σ(f * x) / Σf, where ‘f’ is the frequency of each class interval and ‘x’ is the midpoint of each class interval.


The given data is in the form of class intervals and their corresponding frequencies, representing the number of plants in different houses. This is grouped data. To find the mean number of plants per house, we need to calculate the average. Since the data is grouped, we cannot directly average the numbers. Instead, we use the assumed mean method for grouped data.

Steps to calculate the mean:
1. Find the mid-point (x) of each class interval: The midpoint is calculated by adding the lower and upper limits of the class interval and dividing by 2.
* 0 – 2: (0 + 2) / 2 = 1
* 2 – 4: (2 + 4) / 2 = 3
* 4 – 6: (4 + 6) / 2 = 5
* 6 – 8: (6 + 8) / 2 = 7
* 8 – 10: (8 + 10) / 2 = 9
* 10 – 12: (10 + 12) / 2 = 11
* 12 – 14: (12 + 14) / 2 = 13

2. Multiply the midpoint (x) by the frequency (f) of each class interval (f * x):
* 1 * 1 = 1
* 2 * 3 = 6
* 1 * 5 = 5
* 5 * 7 = 35
* 6 * 9 = 54
* 2 * 11 = 22
* 3 * 13 = 39

3. Sum the frequencies (Σf): This is the total number of houses surveyed.
* Σf = 1 + 2 + 1 + 5 + 6 + 2 + 3 = 20

4. Sum the products of frequency and midpoint (Σ(f * x)):
* Σ(f * x) = 1 + 6 + 5 + 35 + 54 + 22 + 39 = 162

5. Calculate the mean: Mean = Σ(f * x) / Σf
* Mean = 162 / 20 = 8.1

Method used:
I used the direct method for finding the mean of grouped data. This method is suitable when the class intervals are relatively small and the frequencies are not excessively large, making the calculation of f * x manageable. It is a straightforward approach to calculate the average value.

Answer:
The mean number of plants per house is 8.1.

Question:

The question asks for the mean concentration of SO2 in the air from grouped data. To calculate the mean of grouped data, we use the formula: Mean = Σ(f * x) / Σf, where ‘f’ is the frequency of each class interval and ‘x’ is the midpoint of each class interval.


To find the mean concentration of SO2 in the air, we need to perform the following steps:
1. Calculate the midpoint (class mark) for each concentration range. The midpoint is calculated as (lower limit + upper limit) / 2.
2. Multiply the midpoint of each class interval by its corresponding frequency. This gives us the ‘f * x’ value for each interval.
3. Sum up all the ‘f * x’ values.
4. Sum up all the frequencies.
5. Divide the sum of ‘f * x’ values by the sum of frequencies to get the mean concentration.

Let’s apply this to the given data:

Class Interval (concentration of SO2 in ppm) | Frequency (f) | Midpoint (x) | f * x
—————————————–|—————|————–|——-
0.00 − 0.04 | 4 | 0.02 | 0.08
0.04 − 0.08 | 9 | 0.06 | 0.54
0.08 − 0.12 | 9 | 0.10 | 0.90
0.12 − 0.16 | 2 | 0.14 | 0.28
0.16 − 0.20 | 4 | 0.18 | 0.72
0.20 − 0.24 | 2 | 0.22 | 0.44

Sum of frequencies (Σf) = 4 + 9 + 9 + 2 + 4 + 2 = 30
Sum of (f * x) (Σ(f * x)) = 0.08 + 0.54 + 0.90 + 0.28 + 0.72 + 0.44 = 2.96

Mean concentration = Σ(f * x) / Σf = 2.96 / 30

Mean concentration = 0.098666…

Rounding to a suitable number of decimal places (usually 2 or 3 for this type of data), the mean concentration of SO2 in the air is approximately 0.099 ppm.

Question:

To find the mode of a grouped data, we first identify the modal class, which is the class interval with the highest frequency. Then, we use the formula for the mode of grouped data: Mode = L + ((f1 – f0) / (2f1 – f0 – f2)) * h, where L is the lower limit of the modal class, f1 is the frequency of the modal class, f0 is the frequency of the class preceding the modal class, f2 is the frequency of the class succeeding the modal class, and h is the class width.


The given data is in the form of a frequency distribution table. We need to find the mode of this grouped data.
Step 1: Identify the modal class.
The modal class is the class interval with the highest frequency.
Looking at the table:
Runs scored | Number of batsmen (Frequency)
——————————————-
3000 – 4000 | 4
4000 – 5000 | 18
5000 – 6000 | 9
6000 – 7000 | 7
7000 – 8000 | 6
8000 – 9000 | 3
9000 – 10000 | 1
10000 – 11000 | 1
The highest frequency is 18, which corresponds to the class interval 4000 – 5000.
Therefore, the modal class is 4000 – 5000.

Step 2: Identify the values for the mode formula.
L = Lower limit of the modal class = 4000
f1 = Frequency of the modal class = 18
f0 = Frequency of the class preceding the modal class = 4 (frequency of 3000 – 4000)
f2 = Frequency of the class succeeding the modal class = 9 (frequency of 5000 – 6000)
h = Class width = Upper limit – Lower limit = 5000 – 4000 = 1000

Step 3: Apply the mode formula.
Mode = L + ((f1 – f0) / (2f1 – f0 – f2)) * h
Mode = 4000 + ((18 – 4) / (2 * 18 – 4 – 9)) * 1000
Mode = 4000 + ((14) / (36 – 4 – 9)) * 1000
Mode = 4000 + ((14) / (23)) * 1000
Mode = 4000 + (14000 / 23)
Mode = 4000 + 608.6956…
Mode ≈ 4000 + 608.70
Mode ≈ 4608.70

The mode of the data is approximately 4608.70 runs.

Question:

To find the mean of grouped data, we use the formula: Mean = Σ(f*x) / Σf, where f is the frequency of each class interval and x is the midpoint of each class interval.


The question asks for the mean number of days a student was absent. We have the number of days (class intervals) and the number of students (frequencies). To calculate the mean, we need to:

1. Find the midpoint (x) for each class interval. The midpoint is calculated as (lower limit + upper limit) / 2.
2. Multiply the midpoint (x) of each class interval by its corresponding frequency (f). This gives us the product (f*x).
3. Sum all the (f*x) values. This is Σ(f*x).
4. Sum all the frequencies (f). This is Σf, which is already given as 40 students.
5. Divide the sum of (f*x) by the sum of frequencies (Σf) to get the mean.

Step-by-step derivation:
First, create a table with columns for Class Interval, Frequency (f), Midpoint (x), and f*x.

Class Interval | Frequency (f) | Midpoint (x) | f*x
———— | ————- | ———— | —-
0 – 6 | 11 | (0+6)/2 = 3 | 11 * 3 = 33
6 – 10 | 10 | (6+10)/2 = 8 | 10 * 8 = 80
10 – 14 | 7 | (10+14)/2 = 12| 7 * 12 = 84
14 – 20 | 4 | (14+20)/2 = 17| 4 * 17 = 68
20 – 28 | 4 | (20+28)/2 = 24| 4 * 24 = 96
28 – 38 | 3 | (28+38)/2 = 33| 3 * 33 = 99
38 – 40 | 1 | (38+40)/2 = 39| 1 * 39 = 39

Now, calculate the sum of f*x:
Σ(f*x) = 33 + 80 + 84 + 68 + 96 + 99 + 39 = 599

The sum of frequencies is given as Σf = 40.

Finally, calculate the mean:
Mean = Σ(f*x) / Σf
Mean = 599 / 40
Mean = 14.975

Therefore, the mean number of days a student was absent is 14.975.

Question:

To find the mean of grouped data, we use the formula: Mean = Σ(fᵢxᵢ) / Σfᵢ, where fᵢ is the frequency of each class and xᵢ is the midpoint of each class interval.


The table provides the literacy rate in percentages for different groups of cities and the number of cities in each group. This is grouped data. To calculate the mean literacy rate, we need to follow these steps:

1. Find the midpoint (xᵢ) of each class interval: The midpoint is calculated as (lower limit + upper limit) / 2.
* For 45 − 55: x₁ = (45 + 55) / 2 = 50
* For 55 − 65: x₂ = (55 + 65) / 2 = 60
* For 65 − 75: x₃ = (65 + 75) / 2 = 70
* For 75 − 85: x₄ = (75 + 85) / 2 = 80
* For 85 − 95: x₅ = (85 + 95) / 2 = 90

2. Multiply the frequency (fᵢ) of each class by its corresponding midpoint (xᵢ): This gives us the product fᵢxᵢ.
* f₁x₁ = 3 × 50 = 150
* f₂x₂ = 10 × 60 = 600
* f₃x₃ = 11 × 70 = 770
* f₄x₄ = 8 × 80 = 640
* f₅x₅ = 3 × 90 = 270

3. Sum all the products (Σfᵢxᵢ):
* Σfᵢxᵢ = 150 + 600 + 770 + 640 + 270 = 2430

4. Sum all the frequencies (Σfᵢ): This is the total number of cities, which is given as 35.
* Σfᵢ = 3 + 10 + 11 + 8 + 3 = 35

5. Calculate the mean literacy rate:
* Mean = Σ(fᵢxᵢ) / Σfᵢ
* Mean = 2430 / 35

6. Simplify the fraction:
* Mean = 2430 ÷ 35 = 69.42857…

Rounding to a reasonable number of decimal places, the mean literacy rate is approximately 69.43%.

Question:

The mode of a grouped data is the value that occurs most frequently. For grouped data, the mode lies within the modal class, which is the class interval with the highest frequency. The formula to calculate the mode of grouped data is:
Mode = L + [(f1 – f0) / (2f1 – f0 – f2)] * h
where:
L = lower 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 width


1. Identify the modal class: Look for the highest frequency in the given table. The class interval corresponding to this highest frequency is the modal class.
In the given data, the highest frequency is 20, which corresponds to the class interval 40 – 50. Therefore, the modal class is 40 – 50.

2. Determine the values for the mode formula:
L (lower limit of the modal class) = 40
f1 (frequency of the modal class) = 20
f0 (frequency of the class preceding the modal class) = 12 (frequency of 30 – 40)
f2 (frequency of the class succeeding the modal class) = 11 (frequency of 50 – 60)
h (class width) = 10 (calculated as 10 – 0, 20 – 10, etc.)

3. Apply the mode formula:
Mode = 40 + [(20 – 12) / (2 * 20 – 12 – 11)] * 10
Mode = 40 + [8 / (40 – 12 – 11)] * 10
Mode = 40 + [8 / (40 – 23)] * 10
Mode = 40 + [8 / 17] * 10
Mode = 40 + (80 / 17)
Mode = 40 + 4.7058…
Mode ≈ 44.71

Therefore, the mode of the data is approximately 44.71 cars.