Question

The height of females in the United States follows a normal distribution with a mean of 65 inches and a standard deviation of 2 inches.

What height marks the median?

Answer 1

Let us define a random variable X such that:

X: Height of females in the United States.

If a random variable X follows a Normal distribution, then it is denoted as:

, where are the mean and variance of the distribution respectively.

Here,

Thus,

For a normal distribution, we know that

Mean = Median = Mode

Therefore, the median height = mean height = 65 inches.

This can also be solved manually in the following manner:

Let the median height be denoted by k. It is that value which divides the distribution into two equal parts i.e., 50% of the femlaes are shorter than k and the remaining 50% are taller than k.

Therefore, in notations it can be written as:

Now let

z is called the standard normal variate.

So, can be written as

The value of z₁ is obatined from the normal distribution tables and substituted in the equation , which gives

k = 65 inches. Therefore, the median height of females in the United States is 65 inches.