Question

find each measure of spread. round to one decimal place as needed. population standard deviation: type your answer... inter - quartile range: type your answer... based of the shape of the distribution, which measure of spread would be best to describe the distribution. choose your answer... standard deviation, because there are no outliers. iqr, because of the existence of an outlier. 2010 winter olympics gold medal wins by top 20 medal - winning countries number of gold medals won

Explanation:

Step1: Count data points

Count the number of 'x's in the dot - plot to get the data set. Let the data set be \(x_1,x_2,\cdots,x_n\). Here \(n = 20\).

Step2: Calculate mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\). First, list out the values from the dot - plot: \(0\) (2 times), \(1\) (4 times), \(2\) (3 times), \(3\) (1 time), \(4\) (2 times), \(5\) (2 times), \(6\) (2 times), \(9\) (2 times), \(10\) (1 time), \(14\) (1 time). \(\sum_{i=1}^{n}x_i=0\times2 + 1\times4+2\times3 + 3\times1+4\times2+5\times2+6\times2+9\times2+10\times1+14\times1=4 + 6+3+8+10+12+18+10+14=85\), \(\bar{x}=\frac{85}{20}=4.25\).

Step3: Calculate variance

The variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}\).
\((0 - 4.25)^2\times2+(1 - 4.25)^2\times4+(2 - 4.25)^2\times3+(3 - 4.25)^2\times1+(4 - 4.25)^2\times2+(5 - 4.25)^2\times2+(6 - 4.25)^2\times2+(9 - 4.25)^2\times2+(10 - 4.25)^2\times1+(14 - 4.25)^2\times1\)
\(=( - 4.25)^2\times2+( - 3.25)^2\times4+( - 2.25)^2\times3+( - 1.25)^2\times1+( - 0.25)^2\times2+(0.75)^2\times2+(1.75)^2\times2+(4.75)^2\times2+(5.75)^2\times1+(9.75)^2\times1\)
\(=18.0625\times2 + 10.5625\times4+5.0625\times3+1.5625\times1+0.0625\times2+0.5625\times2+3.0625\times2+22.5625\times2+33.0625\times1+95.0625\times1\)
\(=36.125+42.25+15.1875 + 1.5625+0.125+1.125+6.125+45.125+33.0625+95.0625=275.625\)
\(\sigma^{2}=\frac{275.625}{20}=13.78125\).

Step4: Calculate standard deviation

The population standard deviation \(\sigma=\sqrt{\sigma^{2}}=\sqrt{13.78125}\approx3.7\).

Step5: Calculate quartiles

First, order the data: \(0,0,1,1,1,1,2,2,2,3,4,4,5,5,6,6,9,9,10,14\).
Since \(n = 20\), the median is the average of the 10th and 11th ordered values. The 10th value is \(3\) and the 11th value is \(4\), so the median \(Q_2=\frac{3 + 4}{2}=3.5\).
The lower half of the data is \(0,0,1,1,1,1,2,2,2,3\), and the median of the lower half \(Q_1\) (the first - quartile) is the average of the 5th and 6th ordered values in the lower half, \(Q_1=\frac{1+1}{2}=1\).
The upper half of the data is \(4,4,5,5,6,6,9,9,10,14\), and the median of the upper half \(Q_3\) (the third - quartile) is the average of the 5th and 6th ordered values in the upper half, \(Q_3=\frac{6 + 6}{2}=6\).
The inter - quartile range \(IQR=Q_3 - Q_1=6 - 1 = 5\).

Step6: Determine best measure

The distribution has an outlier (the value \(14\)). The inter - quartile range is less affected by outliers than the standard deviation. So the inter - quartile range is the best measure to describe the spread.

Answer:

Population Standard Deviation: \(3.7\)
Inter - Quartile Range: \(5\)
Based on the shape of the distribution, which measure of spread would be best to describe the distribution: IQR, because of the existence of an outlier.