Question

1. when the mean is greater than the median, the data is...

a. skewed left c. bell - shaped / symmetrical
b. skewed right d. uniform

1. the following represents a sample of raw data.

18 21 15 13
find the following.
mean: median:
mode: midrange:
range: what is the deviation for the value of 13?

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a set of data $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 4$, $x_1=18$, $x_2 = 21$, $x_3=15$, $x_4=13$. Then $\sum_{i=1}^{4}x_i=18 + 21+15 + 13=67$. So, $\bar{x}=\frac{67}{4}=16.75$.

Step2: Calculate the median

First, order the data: $13,15,18,21$. Since $n = 4$ (even), the median $M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}$. Here, $x_{2}=15$ and $x_{3}=18$. So, $M=\frac{15 + 18}{2}=16.5$.

Step3: Calculate the mode

The mode is the most frequently - occurring value. In the data set $\{13,15,18,21\}$, each value occurs exactly once. So, there is no mode.

Step4: Calculate the mid - range

The mid - range $MR=\frac{\text{Min}+\text{Max}}{2}$. Here, $\text{Min}=13$ and $\text{Max}=21$. So, $MR=\frac{13 + 21}{2}=17$.

Step5: Calculate the range

The range $R=\text{Max}-\text{Min}$. So, $R=21-13 = 8$.

Step6: Calculate the deviation for the value of 13

The deviation of a value $x$ from the mean $\bar{x}$ is $x-\bar{x}$. For $x = 13$, the deviation is $13-16.75=-3.75$.

Answer:

Mean: $16.75$
Median: $16.5$
Mode: No mode
Midrange: $17$
Range: $8$
Deviation for $13$: $- 3.75$