The I.Q.s of HIM students follow a normal distribution with a mean of 115 and a standard deviation of 10. What does this imply?

In a normal distribution, the mean represents the central point around which the data is distributed, and it is symmetrical. Given that the mean IQ is 115 with a standard deviation of 10, this indicates that most scores cluster around 115, falling within a range defined by the standard deviations.

When considering the statement that 5% will have IQs less than 105, this is based on the properties of the normal distribution. The mean of 115 minus one standard deviation (10) gives us 105. In a standard normal distribution, approximately 68% of data falls within one standard deviation of the mean in both directions (from 105 to 125). We can further parse it to recognize that around 50% of data will fall below the mean of 115, and since the distribution is symmetrical, the area left of 105 represents a smaller percentage.

More specifically, the point 105 lies one standard deviation below the mean. Statistically, about 16% of the data would fall below one standard deviation on the left (lower tail) in a standard normal distribution. Consequently, while 5% falling below a value represents a more specific tail cut-off, the general rule of thumb reinforces that there will be a