Question

directions:

a. for each of the following problems
find the mean, median, mode, range, and standard deviation of the data set.
b. create a box-and-whisker plot for each set of data.
if an answer needs to be rounded, round to the nearest tenth.

1. the data set gives the prices of baseball gloves at dick’s sporting goods.

60, 50, 75, 80, 60, 35, 65

mean\tmedian\tmode\trange\tstandard deviation

min\tq1\tmedian\tq3\tmax

Explanation:

Step1: Organize the data

First, we sort the data set: \( 35, 50, 60, 60, 65, 75, 80 \)

Step2: Calculate the Mean

The mean (\(\bar{x}\)) is the sum of all values divided by the number of values (\(n = 7\)):

$$ \bar{x}=\frac{35 + 50 + 60 + 60 + 65 + 75 + 80}{7}=\frac{425}{7}\approx60.7 $$

Step3: Find the Median

The median is the middle value of the sorted data. Since \(n = 7\) (odd), the median is the 4th value: \(60\)

Step4: Determine the Mode

The mode is the value that appears most frequently. \(60\) appears twice, others once: \(60\)

Step5: Calculate the Range

Range = Max - Min = \(80 - 35 = 45\)

Step6: Compute the Standard Deviation

First, find the squared differences from the mean:

• \((35 - 60.7)^2\approx660.5\)
• \((50 - 60.7)^2\approx114.5\)
• \((60 - 60.7)^2\approx0.5\)
• \((60 - 60.7)^2\approx0.5\)
• \((65 - 60.7)^2\approx18.5\)
• \((75 - 60.7)^2\approx204.5\)
• \((80 - 60.7)^2\approx372.5\)

Sum of squared differences: \(660.5 + 114.5 + 0.5 + 0.5 + 18.5 + 204.5 + 372.5 = 1371.5\)

Variance (\(s^2\)) = \(\frac{\text{Sum of squared differences}}{n - 1}=\frac{1371.5}{6}\approx228.6\)

Standard Deviation (\(s\)) = \(\sqrt{228.6}\approx15.1\)

Step7: Find Min, Q1, Median, Q3, Max for Box - and - Whisker Plot

• Min = \(35\)
• Q1: The median of the lower half (\(35, 50, 60\)) is \(50\)
• Median (already found) = \(60\)
• Q3: The median of the upper half (\(65, 75, 80\)) is \(75\)
• Max = \(80\)

Answer:

Part a:

• Mean: \(\approx60.7\)
• Median: \(60\)
• Mode: \(60\)
• Range: \(45\)
• Standard Deviation: \(\approx15.1\)

Part b (Box - and - Whisker Plot Values):

• Min: \(35\)
• Q1: \(50\)
• Median: \(60\)
• Q3: \(75\)
• Max: \(80\)

To create the box - and - whisker plot, plot a number line, mark Min (\(35\)), Q1 (\(50\)), Median (\(60\)), Q3 (\(75\)), and Max (\(80\)). Draw a box from Q1 to Q3 with a line at the median, and draw whiskers from the box to Min and Max.