To determine the five-number summary and interquartile range for this data set, I'll go through each component.

Data Set

The scores are: 56, 57, 59, 61, 61, 64, 69, 70, 73, 74, 77, 82, 82, 91

Since there are 14 data points, the steps are as follows:

Step 1: Minimum and Maximum

• Minimum: 56
• Maximum: 91

Step 2: Median (Middle Value)

• With 14 values, the median will be the average of the 7th and 8th values (when ordered).
• Median = (69 + 70) / 2 = 69.5

Step 3: Lower Quartile (Q1)

• The lower quartile, Q1, is the median of the first half of the data (first 7 values).
• First half: 56, 57, 59, 61, 61, 64, 69
• Lower Quartile (Q1) = 59

Step 4: Upper Quartile (Q3)

• The upper quartile, Q3, is the median of the second half of the data (last 7 values).
• Second half: 70, 73, 74, 77, 82, 82, 91
• Upper Quartile (Q3) = 77

Step 5: Interquartile Range (IQR)

• The IQR is the difference between Q3 and Q1.
• Interquartile Range (IQR) = Q3 - Q1 = 77 - 59 = 18

Final Summary

Here's the five-number summary and interquartile range for the data set:

• Minimum: 56
• Lower Quartile (Q1): 59
• Median: 69.5
• Upper Quartile (Q3): 77
• Maximum: 91
• Interquartile Range (IQR): 18

Would you like more details on any of these steps or have any questions?

Related Questions

1. How do you calculate the median in a data set with an odd number of values?
2. What does the interquartile range tell you about the data spread?
3. How is the five-number summary helpful in data analysis?
4. How would the five-number summary change if an outlier were added to the data set?
5. Can the median ever be one of the quartiles?

Tip

The interquartile range (IQR) is useful for identifying the spread of the middle 50% of data and is often used to detect outliers.