Descriptive Statistics & Sampling — Honors Math practice questions

Question 1

If the data set $\{x_1, \ldots, x_n\}$ has standard deviation $\sigma$, the standard deviation of $\{2x_1, \ldots, 2x_n\}$ is:

Accepted Answer

$\sigma(c \cdot x) = |c| \sigma(x)$. With $c = 2$: $\sigma(2x) = 2\sigma$.

Answer

$\sigma$

Answer

$\sigma + 2$

Answer

$2\sigma$

Answer

$4\sigma$

Question 2

If the mean of $\{x_1, x_2, \ldots, x_n\}$ is $\bar{x}$, the mean of $\{x_1 + 5, x_2 + 5, \ldots, x_n + 5\}$ is:

Accepted Answer

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

Answer

$\bar{x}$

Answer

$\bar{x} + 5$

Answer

$5\bar{x}$

Answer

$\bar{x} + \dfrac{5}{n}$

Question 3

The first quartile $Q_1$ of $\{2, 4, 5, 7, 8, 11, 14\}$ is:

Accepted Answer

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 $Q_1 = 4$.

Answer

$3$

Answer

$4$

Answer

$5$

Answer

$7$

Question 4

The variance of $\{2, 4, 4, 4, 5, 5, 7, 9\}$ (population) is:

Accepted Answer

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

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$5$

Question 5

The median of $\{4, 1, 3, 8, 5, 6\}$ is:

Accepted Answer

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

Answer

$4$

Answer

$4.5$

Answer

$5$

Answer

$5.5$

Question 6

The range of $\{12, 7, 22, 9, 18, 4\}$ is:

Accepted Answer

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

Answer

$4$

Answer

$9$

Answer

$18$

Answer

$22$

Question 7

A class has $20$ students. Their math grades are: $5$ students got $7$, $8$ got $8$, $7$ got $9$. The mean grade is:

Accepted Answer

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

Answer

$7.8$

Answer

$8.0$

Answer

$8.1$

Answer

$8.5$

Question 8

A football coach records the number of goals scored by a player in each of $5$ matches:

$$3,\ 3,\ 4,\ 6,\ 9$$

Work out the **mean** number of goals per match.

Accepted Answer

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.)

Answer

$5$

Answer

$4$

Answer

$3$

Answer

$6$

Question 9

The median of $\{3, 7, 1, 9, 5\}$ is:

Accepted Answer

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

Answer

$3$

Answer

$5$

Answer

$7$

Answer

$9$

Question 10

The mode of $\{2, 3, 5, 5, 7, 8, 5, 9\}$ is:

Accepted Answer

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

Answer

$2$

Answer

$3$

Answer

$5$

Answer

$7$

Descriptive Statistics & Sampling practice questions — Honors Math

Problems & worked solutions

Problem 1 — Descriptive Statistics & Sampling

Problem 2 — Descriptive Statistics & Sampling

Problem 3 — Descriptive Statistics & Sampling

Problem 4 — Descriptive Statistics & Sampling

Problem 5 — Descriptive Statistics & Sampling

Problem 6 — Descriptive Statistics & Sampling

Problem 7 — Descriptive Statistics & Sampling

Problem 8 — Descriptive Statistics & Sampling

Problem 9 — Descriptive Statistics & Sampling

Problem 10 — Descriptive Statistics & Sampling