Data Handling Questions with Solution
Q1. The following dataset represents the scores obtained by students in a mathematics exam: [75, 80, 85, 90, 85, 70, 80, 85, 90, 95]. Calculate the mean, median, and mode of the dataset.
Solution:
Mean = (75 + 80 + 85 + 90 + 85 + 70 + 80 + 85 + 90 + 95) / 10 = 855 / 10 = 85.5
Median = (85 + 85) / 2 = 85
Mode = 85
Q2. Compute the range, variance, and standard deviation for the following dataset: [10, 15, 20, 25, 30]
Solution:
Range = Maximum value – Minimum value = 30 – 10 = 20
Mean = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20
Variance = [(10 – 20)2 + (15 – 20)2 + (20 – 20)2 + (25 – 20)2 + (30 – 20)2] / 5
= (100 + 25 + 0 + 25 + 100) / 5 = 250 / 5 = 50
Standard Deviation = √Variance = √50 ≈ 7.07
Q3. Calculate the Pearson correlation coefficient (r) for the following dataset:
X: [10, 15, 20, 25, 30]
Y: [20, 25, 30, 35, 40]
Solution:
Mean of X = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20
Mean of Y = (20 + 25 + 30 + 35 + 40) / 5 = 150 / 5 = 30
Σ((x – x̄)(y – ȳ)) = (10 – 20)(20 – 30) + (15 – 20)(25 – 30) + (20 – 20)(30 – 30) + (25 – 20)(35 – 30) + (30 – 20)(40 – 30)
= (-10 × -10) + (-5 × -5) + (0 × 0) + (5 × 5) + (10 × 10)
= 100 + 25 + 0 + 25 + 100 = 250
Σ(x – x̄)2 = (10 – 20)2 + (15 – 20)2 + (20 – 20)2 + (25 – 20)2 + (30 – 20)2
= 100 + 25 + 0 + 25 + 100 = 250
Σ(y – ȳ)2 = (20 – 30)2 + (25 – 30)2 + (30 – 30)2 + (35 – 30)2 + (40 – 30)2
= 100 + 25 + 0 + 25 + 100 = 250
r = Σ((x – x̄)(y – ȳ)) / √(Σ(x – x̄)2 × Σ(y – ȳ)2)
= 250 / √(250 × 250) = 250 / 250 = 1 ×
Q4. Perform a t-test for the given dataset to test the hypothesis that the mean is 20:
Dataset: [18, 19, 21, 22, 20, 23, 17, 20, 19, 20]
(Assuming a significance level of 0.05)
Solution:
Mean = (18 + 19 + 21 + 22 + 20 + 23 + 17 + 20 + 19 + 20) / 10 = 199 / 10 = 19.9
Standard Deviation = √[(Σ(x – x̄)2) / (n – 1)] = √[(16.9 + 9.6 + 0.1 + 4.1 + 0.1 + 9.6 + 5.6 + 0.1 + 0.1 + 0.1) / 9]
= √(45.2 / 9) = √5.022 ≈ 2.24
t = (X̄ – μ) / (s / √n) = (19.9 – 20) / (2.24 / √10) ≈ -0.224
Degrees of Freedom (df) = n – 1 = 10 – 1 = 9
Critical t-value for df = 9 at α = 0.05 (two-tailed) is approximately ±2.262
Since |-0.224| < 2.262, we fail to reject the null hypothesis.
Q5. The heights (in inches) of a sample of 5 students are as follows: 65, 68, 70, 63, 72. Calculate the mean height of the students.
Solution :
Mean = (65 + 68 + 70 + 63 + 72) / 5
Mean = 338 / 5
Mean = 67.6 inches
Q6. Calculate the variance of the following dataset: 5, 8, 10, 12, 15.
Solution :
Mean = (5 + 8 + 10 + 12 + 15) / 5
Mean = 50 / 5
Mean = 10.
Now, calculate the squared deviations from the mean:
(5 – 10)2 = 25
(8 – 10)2 = 4
(10 – 10)2 = 0
(12 – 10)2 = 4
(15 – 10)2 = 25
Variance = (25 + 4 + 0 + 4 + 25) / 5
Variance = 58 / 5
Variance = 11.6.
Q7. What is the correlation coefficient if the covariance between two variables X and Y is 50, the standard deviation of X is 5, and the standard deviation of Y is 10?
Solution :
Correlation coefficient (r) = Covariance / (Standard deviation of X × Standard deviation of Y)
r = 50 / (5 × 10)
r = 50 / 50
r = 1
Q8. Perform a t-test with the following data: sample mean = 65, population mean = 60, sample standard deviation = 8, sample size = 25. Assume a significance level of 0.05.
Solution:
t = (X̄ – μ) / (s / √n)
t = (65 – 60) / (8 / √25)
t = 5 / (8 / 5)
t = 5 / 1.6
t ≈ 3.125.
With a significance level of 0.05 and 24 degrees of freedom (n – 1), the critical t-value is approximately 2.064. Since 3.125 > 2.064, we reject the null hypothesis.
Q9. Calculate the median of the following dataset: 12, 15, 18, 20, 22, 25, 28, 30.
Solution:
Since there are 8 data points, the median is the average of the 4th and 5th terms.
Median = (20 + 22) / 2
Median = 21.
Question :
Find the range of the following dataset: 10, 15, 20, 25, 30.
Solution :
Range = Maximum value – Minimum value
Range = 30 – 10
Range = 20
Practice Questions on Data Handling
Data handling refers to the process of managing and manipulating data. In this article, we will learn how to solve questions based on data handling. This article provides practice questions based on data handling.