Scores on the LSAT are approximately normally distributed. In fact, published reports indicate that approximately 40 percent of all test takers score at or between 145 and 155, and about 70 percent score at or between 140 and 160. The full range of scores is 60 pts (120–180). Using your knowledge of normal distributions, estimate the mean, median, and the standard deviation of the LSAT.
To estimate the mean, median, and standard deviation of the LSAT scores, we can use the information given about the percentage of test takers scoring within certain ranges.
First, let's find the mean:
The mean is the average score, and since the LSAT scores are approximately normally distributed, the mean will be the midpoint of the score range. The range of scores is 60 points (from 120 to 180).
So, the mean would be (120 + 180) / 2 = 150.
Next, let's find the standard deviation:
The standard deviation measures the spread or variability of the scores around the mean. However, we don't have direct information about the standard deviation.
We know that about 70% of test takers score between 140 and 160. In a standard normal distribution, this corresponds to a range of approximately 1 standard deviation on each side of the mean.
Since 140 and 160 are roughly 1 standard deviation away from the mean, we can estimate the standard deviation as half the range, which is (160 - 140) / 2 = 10.
Finally, let's find the median:
The median is the score at which 50% of the test takers scored below and 50% scored above. From the given information, we know that approximately 40% scored at or between 145 and 155.
Since this range is centered around the mean, the median should be close to the mean, which we estimated as 150. So, we can estimate the median as 150.
To summarize:
- Estimated mean: 150
- Estimated median: 150
- Estimated standard deviation: 10