Statistics: center and spread

High School Math 1 · Unit 17 · descriptive-stats-spread · Teacher edition

descriptive-stats-spread

Name ______________________________Date ______________

1.
Find the mean of $4,\ 7,\ 7,\ 10,\ 12$ .
Answer: ______________
2.
Find the population standard deviation $\sigma$ of $5,\ 7,\ 8,\ 9,\ 11$ .
Answer: ______________
3.
Eight runners' times, in minutes: $14,\ 15,\ 16,\ 18,\ 20,\ 21,\ 22,\ 40$ . Using the $1.5 \times IQR$ rule, what is the upper fence (the value above which a time counts as an outlier)?
Answer: ______________
4.
Six quiz scores are $7,\ 9,\ 9,\ 10,\ 12,\ 13$ . Their mean is $10$ . What is the population standard deviation $\sigma$ ?
1. (A)
  $2$
2. (B)
  $4$
3. (C)
  $\sqrt{24}$
4. (D)
  $24$
5.
A data set has mean $40$ and standard deviation $5$ . Every value in the set is doubled. What are the mean and standard deviation of the new data set?
1. (A)
  mean $80$ , standard deviation $5$
2. (B)
  mean $40$ , standard deviation $10$
3. (C)
  mean $80$ , standard deviation $20$
4. (D)
  mean $80$ , standard deviation $10$

Answer key — Statistics: center and spread

1.
8descriptive-stats-spread-01
$\bar x = \dfrac{4 + 7 + 7 + 10 + 12}{5} = \dfrac{40}{5} = \mathbf{8}$ .
2.
2descriptive-stats-spread-02
Mean $8$ . Squared deviations sum to $9 + 1 + 0 + 1 + 9 = 20$ . Variance $= \frac{20}{5} = 4$ , so $\sigma = \sqrt{4} = \mathbf{2}$ .
3.
30.5descriptive-stats-spread-03
$Q_1 = 15.5$ , $Q_3 = 21.5$ , $IQR = 6$ . Upper fence $= 21.5 + 1.5(6) = 21.5 + 9 = \mathbf{30.5}$ minutes. The $40$ -minute time is above the fence, so it is an outlier.
4.
(A)
$2$
descriptive-stats-spread-04
$\sum (x_i - \bar x)^2 = 24$ ; variance $= \frac{24}{6} = 4$ ; $\sigma = \sqrt{4} = \mathbf{2}$ . ( $4$ is the variance — the square root was skipped; $24$ is the sum of squares; $\sqrt{24}$ skips the division by $n$ .)
5.
(D)
mean $80$ , standard deviation $10$
descriptive-stats-spread-05
New mean $= 2 \cdot 40 = 80$ ; new standard deviation $= 2 \cdot 5 = 10$ . Answer: mean 80, standard deviation 10. (“SD stays $5$ ” is the rule for adding a constant; SD $20$ applies the factor $4$ that belongs to the variance.)