the dotplot shows the difference (highway - city) in epa mileage ratings for each of 21 model year 2020…

Explanation:

Step1: Organize data from dot - plot

First, list out the data values from the dot - plot. Let (x_1,x_2,\cdots,x_{21}) be the difference in gas mileage values.

Step2: Calculate the median (Q2)

Arrange the 21 data values in ascending order. Since (n = 21), the median is the (\left(\frac{n + 1}{2}\right))-th value. (\frac{21+1}{2}=11) - th value.

Step3: Calculate Q1 and Q3

The lower half of the data has (n_1=10) values. The median of the lower half (Q1) is the (\left(\frac{10 + 1}{2}\right)=5.5) - th value (average of 5 - th and 6 - th ordered values). The upper half of the data has (n_2 = 10) values. The median of the upper half (Q3) is the (\left(11+\frac{10 + 1}{2}\right)=16.5) - th value (average of 16 - th and 17 - th ordered values). Then (IQR=Q3 - Q1).

Step4: Calculate the mean (\bar{x})

(\bar{x}=\frac{\sum_{i = 1}^{21}x_i}{21})

Step5: Calculate the variance (s^2)

(s^2=\frac{\sum_{i = 1}^{21}(x_i-\bar{x})^2}{21 - 1})

Step6: Calculate the standard deviation (s)

(s=\sqrt{s^2}), and round to 2 decimal places.

Let's assume the data values from the dot - plot are: (- 3,7,7,8,8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,13,14)

The ordered data set has (n = 21) values. The median (Q2) is the 11 - th value, which is 10. The lower half of the data: (-3,7,7,8,8,8,9,9,9,9) Q1 is the average of the 5 - th and 6 - th values (\frac{8 + 8}{2}=8) The upper half of the data: (10,10,10,10,10,11,11,11,13,14) Q3 is the average of the 16 - th and 17 - th values (\frac{11+11}{2}=11) (IQR=Q3 - Q1=11 - 8 = 3)

The mean (\bar{x}=\frac{-3+7\times2 + 8\times3+9\times4+10\times6+11\times3+13+14}{21}=\frac{- 3+14 + 24+36+60+33+13+14}{21}=\frac{181}{21}\approx8.62)

((x_1-\bar{x})^2, (x_2-\bar{x})^2,\cdots) are calculated as follows: ((-3 - 8.62)^2=(-11.62)^2 = 135.0244), ((7-8.62)^2=(-1.62)^2 = 2.6244) and so on. (s^2=\frac{\sum_{i = 1}^{21}(x_i - 8.62)^2}{20}) (\sum_{i = 1}^{21}(x_i - 8.62)^2=135.0244+2\times2.6244+3\times(8 - 8.62)^2+4\times(9 - 8.62)^2+6\times(10 - 8.62)^2+3\times(11 - 8.62)^2+(13 - 8.62)^2+(14 - 8.62)^2) (=135.0244 + 5.2488+3\times0.3844+4\times0.1444+6\times1.9044+3\times5.6644+19.1844+28.9444) (=135.0244+5.2488 + 1.1532+0.5776+11.4264+16.9932+19.1844+28.9444) (=218.5524) (s^2=\frac{218.5524}{20}=10.92762) (s=\sqrt{10.92762}\approx3.31)

Answer:

(IQR = 3) Standard deviation (=3.31)