The Mean (X̅) of given normally distributed scores is 40. Arrange the following score intervals in ascending order on the basis of frequency of their classes.  

A. 30 - 50

B. X̅ to 55

C. 30 to X̅

D. 10 to X̅

Choose the correct answer from the options given below: 

The correct answer is: C, B, A, D

Key Points

• Normal Distribution Basics:
  • It is symmetric about the mean, with the highest frequency occurring at the mean score.
  • The frequency decreases as we move away from the mean in either direction.
  • The total area under the normal curve equals 1, representing total probability or total frequency.
• Interpretation of Intervals Relative to the Mean (40):
  • C (30 to X̅): This interval ranges from 30 up to the mean 40, covering a 10-unit range just below the mean. Since it is close to the mean, it contains relatively high frequency.
  • B (X̅ to 55): This interval goes from 40 to 55, covering 15 units just above the mean. It contains slightly less frequency compared to C, as it moves further from the mean.
  • A (30 to 50): This interval spans from 30 to 50, covering 20 units around the mean, including parts below and above it. It includes frequencies from both sides of the mean, thus has a moderate total frequency.
  • D (10 to X̅): This is the largest interval extending farthest below the mean (10 to 40), covering 30 units. However, since it stretches further away from the mean, the average frequency per unit is lower, and the overall frequency is less concentrated.
• Frequency Ordering Logic:
  • The highest frequencies are near the mean; intervals closer to the mean have higher frequency densities.
  • Intervals extending farther from the mean have lower frequency densities due to the shape of the normal curve.
  • Therefore, among the given intervals, the one closest and smallest around the mean (30 to 40) has the highest frequency, and the widest interval extending farthest (10 to 40) has the lowest.

Additional Information

• In a normal distribution, approximately 68% of data lies within one standard deviation from the mean. Although the standard deviation is not given, the frequency concept still follows the proximity principle: scores nearer the mean have higher occurrence.
• Interval C (30 to 40) is closest to the mean on the lower side and therefore will contain the highest frequency among the intervals listed.
• Interval B (40 to 55) moves away from the mean but still within a moderate range, so frequency is somewhat less than C but more than the larger intervals.
• Interval A (30 to 50) includes parts of both sides of the mean but is broader and thus has a moderate frequency, less than intervals closer to the mean.
• Interval D (10 to 40), despite being large, extends farthest from the mean, where frequencies tail off, resulting in the lowest frequency of all intervals.
• Thus, ordering frequencies in ascending order as C, B, A, D aligns with the characteristics of the normal distribution curve.