Descriptive Statistics & Sampling practice questions — Honors Math

$\sigma (c\cdot x)=\mathrm{\mid }c\mathrm{\mid }\sigma (x)$. With $c=2$: $\sigma (2x)=2\sigma$.

${\displaystyle \frac{1}{n}}\sum (x_i+5)={\displaystyle \frac{1}{n}}\sum x_i+5=\bar{x}+5$.

Sorted (already): $\{2,4,5,7,8,11,14\}$. Median is $7$ (position 4). Lower half: $\{2,4,5\}$. Median of lower half: $4$. So $Q_1=4$.

Sum $=40$, $n=8$, so $\bar{x}=5$. Squared deviations: $9,1,1,1,0,0,4,16$ — sum $=32$. Variance $={\displaystyle \frac{32}{8}}=4$.

Sorted: $\{1,3,4,5,6,8\}$. The two middle values are $4$ and $5$. Median $={\displaystyle \frac{4+5}{2}}=4.5$.

Max $=22$, min $=4$, so range $=22-4=18$.

Sum $=5\cdot 7+8\cdot 8+7\cdot 9=35+64+63=162$. Mean $={\displaystyle \frac{162}{20}}=8.1$.

Add up all five values: $$3+3+4+6+9=25$$ Divide the total by the number of matches: $$\frac{25}{5}=5$$ So the mean is $5$ goals per match. (Here $4$ is the median, $3$ is the mode, and $6$ is the range.)

Sorted: $\{1,3,5,7,9\}$. The middle value (3rd of 5) is $5$.

The value appearing most often is $5$ (three times).