# Cumulative Distribution Function (CDF) and Means of Central Tendency

Using the graph of the cumulative distributive function below, find the:

(a) mean

(b) median

(c) mode

(d) midrange

(e) third quartile for the random variable

I really just want help with parts (b) and (e).

I started this question by creating a table from the graph and then back calculating the probability function:

$$\begin{array}{l|l} \hline X & 3 & 4 & 6 & 6.5 & 7 & 7.5 \\ \hline F(x) & 0 & 0.2 & 0.4 & 0.8 & 0.9 & 1 \\ \hline Pr(X) & 0 & 0.2 & 0.2 & 0.4 & 0.1 & 0.1 \\ \hline \end{array}$$

(a) $\mu_X=(4)(0.2) + 6(0.2) + 6.5(0.4) + 7(0.1) = 7.5(0.1) = 6.05$

(b) I'm confused on how to calculate the median from the table. I know the median is the center value of a data set or the 50th percentile ($Q_2$).

To calculate percentiles:

If $x_1, x_2, \cdots, x_n$ are $n$ data points arranged in ascending order, then $x_i$ corresponds to the $\left(100 \cdot \frac{i}{n+1}\right)^{th}$ percentile.

So this is what I did: $Q_2 = 50 = 100 \cdot \frac{i}{11}\tag{1}$

I set the $n=10$ because I the probabilities are all in tenths, so I thought you could imagine there are $10$ possible outcomes. So you could say there are two $4$s and $6$s, four $6.5$'s, and one $7$ and one $7.5$.

Since there is an even number of data points (namely $10$), the median will be the mean of the $5$th and $6$th data point. Since $6.5$ is the $5$th, $6$th, $7$th, and $8$th data point, the mean is $6.5$. Is this best or correct approach?

(c) mode = $6.5$

(d) midrange = $(4+7.5)/2 = 5.75$

(e) Not getting this one.

$Q_3: 75 = 100 \cdot \frac{i}{11}$

Solving, $i=8.25$. This is just the index. Now, we have to interpolate to get the value for the $8.25$ index:

$\cfrac{8.25 - 8}{x-6.5} = \cfrac{9-8}{7-6.5}$

$\implies x=6.625$ which is wrong according to the book..... This should be $6.5$. I checked the errata but nothing there. Am I getting the wrong answer? Also , is there a faster way of doing this?

To do this method you have to manually count and find which value of $X$ belongs to the index $8$ and $9$ which can be time consuming if you have a table with $36$ data points. Is there a faster way?

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