Descriptive Statistics
Consider the observations 2, 4, 6, 8, and 10 taken from an experiment, and from these five data points determine the variance of the entire collection.
Select the correct option:
Solution
8
Variance measures the average squared deviation of observations from their mean, quantifying spread, and for n values it is σ² = (1/n)Σ(xᵢ − x̄)². This is a core descriptive-statistics computation routinely tested at JEE Advanced. The mean of 2, 4, 6, 8, 10 is (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6. The deviations from 6 are −4, −2, 0, 2, 4, whose squares are 16, 4, 0, 4, 16, summing to 40. Dividing by n = 5 gives variance 40/5 = 8. Option 10 results from dividing the sum of squared deviations by n − 1 = 4 only partially or mis-summing. Option 6 confuses the variance with the mean. Option 40 reports the total sum of squared deviations without dividing by n. The method follows directly from the population-variance definition, equivalently σ² = (1/n)Σxᵢ² − x̄². Plausibility check: the data are symmetric about 6 and the standard deviation √8 ≈ 2.83 falls comfortably within the range 8 of the dataset, confirming a reasonable spread.
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About This Question
- Subject
- mathematics
- Chapter
- statistics and probability
- Topic
- descriptive statistics
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
8
Variance measures the average squared deviation of observations from their mean, quantifying spread, and for n values it is σ² = (1/n)Σ(xᵢ − x̄)². This is a core descriptive-statistics computation routinely tested at JEE Advanced. The mean of 2, 4, 6, 8, 10 is (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6. The deviations from 6 are −4, −2, 0, 2, 4, whose squares are 16, 4, 0, 4, 16, summing to 40. Dividing by n = 5 gives variance 40/5 = 8. Option 10 results from dividing the sum of squared deviations by n − 1 = 4 only partially or mis-summing. Option 6 confuses the variance with the mean. Option 40 reports the total sum of squared deviations without dividing by n. The method follows directly from the population-variance definition, equivalently σ² = (1/n)Σxᵢ² − x̄². Plausibility check: the data are symmetric about 6 and the standard deviation √8 ≈ 2.83 falls comfortably within the range 8 of the dataset, confirming a reasonable spread.
This easy difficulty mathematics question is from the chapter statistics and probability, covering the topic of descriptive statistics. It appeared in the 2025 exam.
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