A set of 500 values has a normal distribution. The mean of the data is 24 and the standard deviation is 2. Find the range about the mean which includes 50% of the data.

To find the range about the mean which includes 50% of the data, we will use the concept of the standard deviation.

In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.

Since we want to find the range which includes 50% of the data, we need to find the number of standard deviations that corresponds to 50%.

To do this, we'll use the concept of the cumulative distribution function (CDF) of the normal distribution. We want to find the z-score that corresponds to the cumulative area of 0.25 on each side of the mean.

To find the z-score, we can use a standard normal distribution table or a statistical software. The z-score corresponding to a cumulative area of 0.25 on each side of the mean is approximately ±0.674.

Now, we can calculate the range that includes 50% of the data:

Lower Range = Mean - (z-score * standard deviation)
= 24 - (0.674 * 2)
= 24 - 1.348
= 22.652

Upper Range = Mean + (z-score * standard deviation)
= 24 + (0.674 * 2)
= 24 + 1.348
= 25.348

Therefore, the range about the mean which includes 50% of the data is approximately 22.652 to 25.348.