OMPT descriptive statistics practice: lesson and 10 problems

Answer: 9

The sum is $4+7+9+10+15=45$, and dividing by the 5 observations gives a mean of $9$ cm. As a check, the mean here equals the middle value, though that is coincidence, not a rule.

Answer: B — $8.5$

With an even number of ordered observations, the median is the mean of the two middle ones: $\frac{8+9}{2}=8.5$.

Answer: 2.24

The mean is $5$. The squared deviations are $9,1,1,9$, summing to 20. Dividing by $n=4$ gives a variance of 5, so the standard deviation is $\sqrt{5}\approx 2.24$.

Answer: B — Mean increases by 5; standard deviation is unchanged

Shifting all data by the same amount drags the mean along by exactly that amount. The spread, however, is about distances between values, and those distances are untouched by a shift — so the standard deviation stays the same. Multiplying by a constant is the operation that scales the spread.

Answer: Step 3

Steps 1 and 2 are right: with seven values the median is the fourth, and the quartiles are the medians of the lower half $(1,4,6)$ and upper half $(9,11,15)$. Step 3 computes the range, not the interquartile range. The IQR is $Q_3-Q_1=11-4=7$, the width of the middle half of the data.

Answer: 7.5

This is a weighted mean: $0.4\cdot 6.0+0.6\cdot 8.5=2.4+5.1=7.5$. The result sits closer to 8.5 than to 6.0 because homework carries the heavier weight.

Answer: 3.1

I weight each value by its frequency before dividing: total $2\cdot 3+3\cdot 5+5\cdot 2=6+15+10=31$, over $3+5+2=10$ households, so the mean is $\frac{31}{10}=3.1$. The result sits between 2 and 5 and closest to 3, the most frequent value, which is what I would expect.

Answer: A — $10.5$

Eight values split into a lower half $3,5,7,8$ and an upper half $12,13,20,21$. $Q_1$ is the median of the lower half, $\frac{5+7}{2}=6$; $Q_3$ is the median of the upper half, $\frac{13+20}{2}=16.5$. So $\text{IQR}=16.5-6=10.5$. For comparison the overall median is $\frac{8+12}{2}=10$ and the range is $21-3=18$, neither of which is what was asked.

Answer: 36

Subtracting 2 shifts every value equally and does nothing to the spread. Multiplying by 1.5 multiplies the standard deviation by 1.5 and therefore the variance by $1.5^2=2.25$. New variance: $2.25\cdot 16=36$. Checking through the standard deviation: it goes from 4 to 6, and $6^2=36$.

Answer: 2.94

First the missing score: the six scores total $6\cdot 9=54$, the five known ones total $4+7+8+11+12=42$, so the sixth is $54-42=12$. Deviations from the mean 9: $-5,-2,-1,2,3,3$, which sum to zero. Squares: $25,4,1,4,9,9$, total 52. Variance $\frac{52}{6}\approx 8.667$, standard deviation $\sqrt{8.667}\approx 2.94$. The values sit 1 to 5 from the mean, so 2.94 is plausible.