In a large random data set following normal distribution, the ratio (%) of…
2010
In a large random data set following normal distribution, the ratio (%) of number of data points which are in the range of (mean ± standard deviation) to the total number of data points, is
Answer: B. ~ 67% — ConceptFor a normal distribution, the probability within a symmetric interval around the mean is found from the standard normal cumulative distribution…
- A.
~ 50%
- B.
~ 67%
- C.
~ 97%
- D.
~ 47%
Show answer & explanation
Correct answer: B
Concept
For a normal distribution, the probability within a symmetric interval around the mean is found from the standard normal cumulative distribution function.
The empirical 68–95–99.7 rule states that about 68.27% of observations lie within one standard deviation of the mean.
Application
The required interval is from μ − σ to μ + σ.
After standardization, its probability is P(−1 ≤ Z ≤ 1) = Φ(1) − Φ(−1).
Using Φ(1) ≈ 0.8413 and Φ(−1) ≈ 0.1587 gives 0.8413 − 0.1587 = 0.6826, or about 68%.
Cross-check
By symmetry, the area from the mean to one standard deviation is about 34.13% on each side; doubling it gives about 68.26%.
Among the offered values, ~67% is the intended nearest approximation.