Question

below are iq scores from 30 randomly selected adults. { 77, 80, 84, 85, 86, 88, 93, 95, 96, 100, 104, 104, 105, 107, 107, 107, 108, 108, 108, 108, 110, 111, 112, 113, 114, 114, 118, 120, 132, 137 }. first, give the mean of the data set. next, give the median of the data set. now give the mode of the data set. if there is more than one, write them in order, separated by commas. finally, give the midrange of the data set. given the relationship between the mean and median above, what shape is the distribution likely to be? the distribution will be roughly symmetric. the distribution will probably be skewed to the left. the distribution will probably be skewed to the right.

Explanation:

Step1: Recall mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 30$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{30}x_{i}=77 + 80+84 + 85+86+88+93+95+96+100+104+104+105+107+107+107+108+108+108+108+110+111+112+113+114+114+118+120+132+137=3108$
$\bar{x}=\frac{3108}{30}=103.6$

Step2: Recall median formula

Since $n = 30$ (an even number), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered data - points. The 15th and 16th ordered data - points are both 107, so the median $M=\frac{107 + 107}{2}=107$

Step3: Recall mode concept

The mode is the data - point that appears most frequently. 108 appears 4 times, more frequently than any other number, so the mode is 108.

Step4: Recall mid - range formula

The mid - range is $\frac{\text{min}+\text{max}}{2}$, where $\text{min}=77$ and $\text{max}=137$. So the mid - range $=\frac{77 + 137}{2}=\frac{214}{2}=107$

Step5: Analyze distribution shape

When the mean ($\bar{x}=103.6$) is less than the median ($M = 107$), the distribution is probably skewed to the left. This is because in a left - skewed distribution, the tail on the left side pulls the mean in that direction.

Answer:

Part 1: 103.6
Part 2: 107
Part 3: 108
Part 4: 107
Part 5: The distribution will probably be skewed to the left.