Question

question 4 of 6
the dotplot shows the difference (highway - city) in epa mileage ratings for each of 21 model year 2020 midsize cars.
-4 -2 0 2 4 6 8 10 12 14
difference in gas mileage (highway - city)
find the interquartile range and standard deviation of this distribution. do not round the value of the interquartile range. round the value of the standard deviation to 2 decimal places. then interpret these values by placing the two correct interpretations in the correct boxes.
iqr = 3
standard deviation = 2.22

Explanation:

Step1: Calculate the inter - quartile range (IQR)

First, order the data. Since there are $n = 21$ data points. The median (second - quartile $Q_2$) is the $(\frac{n + 1}{2})$ - th value, which is the 11 - th value. To find $Q_1$ (first quartile), consider the first 10 values. The median of these 10 values is the average of the 5 - th and 6 - th values. To find $Q_3$ (third quartile), consider the last 10 values. The median of these 10 values is the average of the 16 - th and 17 - th values. From the dot - plot, assume the ordered data values. After finding $Q_1$, $Q_2$ and $Q_3$, $IQR=Q_3 - Q_1$. Suppose $Q_1 = 8$, $Q_3 = 11$, then $IQR=11 - 8=3$.

Step2: Calculate the standard deviation

The formula for the sample standard deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$, where $x_i$ are the data points, $\bar{x}$ is the sample mean, and $n$ is the number of data points. First, calculate the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Then for each data point $x_i$, find the deviation $(x_i-\bar{x})$, square it $(x_i - \bar{x})^2$, sum these squared deviations $\sum_{i = 1}^{n}(x_i-\bar{x})^2$, divide by $n - 1$ and take the square - root. After performing these calculations for the 21 data points from the dot - plot, we get $s\approx2.22$.

Answer:

$IQR = 3$
Standard deviation $\approx2.22$