Finding the Five-Number Summary

Find the five-number summary for this data set: 12, 18, 22, 25, 28, 30, 35, 42, 45

Order the data from least to greatest: Already ordered: 12, 18, 22, 25, 28, 30, 35, 42, 45 = 9 values in order

Find the minimum and maximum: Minimum = 12, Maximum = 45 = Min = 12, Max = 45

Find the median (middle value): With 9 values, the median is the 5th value: 28 = Median = 28

Find Q1 (median of lower half): Lower half: 12, 18, 22, 25. Q1 = (18 + 22) / 2 = 20 = Q1 = 20

Find Q3 (median of upper half): Upper half: 30, 35, 42, 45. Q3 = (35 + 42) / 2 = 38.5 = Q3 = 38.5

Answer: Five-number summary: Min = 12, Q1 = 20, Median = 28, Q3 = 38.5, Max = 45

Calculating the Interquartile Range (IQR)

Using the five-number summary from Example 1, calculate the IQR and determine if there are any outliers.

Recall Q1 and Q3: Q1 = 20, Q3 = 38.5 = Quartiles identified

Calculate IQR = Q3 - Q1: $38.5 - 20 = 18.5$ = IQR = 18.5

Calculate lower fence: $Q1 - 1.5 \times IQR = 20 - 1.5 \times 18.5 = 20 - 27.75 = -7.75$ = Lower fence = -7.75

Calculate upper fence: $Q3 + 1.5 \times IQR = 38.5 + 1.5 \times 18.5 = 38.5 + 27.75 = 66.25$ = Upper fence = 66.25

Check for outliers: All values (12 to 45) fall within [-7.75, 66.25] = No outliers

Answer: IQR = 18.5. No outliers exist because all data points fall within the fences.

Interpreting a Box Plot

A box plot shows: Min = 52, Q1 = 65, Median = 72, Q3 = 80, Max = 95, with one outlier at 38. What can you conclude about this test score data?

Identify the center: Median = 72, meaning half the students scored above 72 and half below = Typical score around 72

Analyze the spread: IQR = 80 - 65 = 15. The middle 50% of scores span 15 points = Moderate spread

Compare box sections: 65 to 72 = 7 points, 72 to 80 = 8 points. Nearly symmetric = Fairly balanced distribution

Interpret the outlier: Score of 38 is far below the lower fence. This student performed unusually poorly = One unusually low score

Summarize findings: Most students scored between 65-80, with a typical score of 72 = Good overall performance

Answer: Most students performed well (median 72, middle 50% between 65-80). One student scored unusually low (38). The distribution is fairly symmetric with moderate spread.

Mistake: Including the median when finding Q1 and Q3

Why: When you have an odd number of data points, the median is a single value. When finding Q1 and Q3, you should NOT include this median value in either half.

Correct: Split the data at the median, leaving it out of both halves, then find the median of each half separately.

Mistake: Thinking a longer whisker means more data points

Why: Whisker length shows the range of data, not the amount. A long whisker might represent just a few spread-out values.

Correct: Each section of a box plot (each quartile) contains 25% of the data, regardless of how wide or narrow it appears.

Mistake: Confusing median with mean

Why: The line inside the box shows the median (middle value), not the mean (average). These can be very different, especially with outliers.

Correct: Box plots use the median because it's not affected by extreme values, making it a better measure of the typical value.

Comparing Sports Performance

Coaches use box plots to compare player statistics and identify consistent performers.

Two basketball players' points per game are shown in box plots. Player A: median 18, IQR 4. Player B: median 20, IQR 12. Player B scores higher on average but Player A is more consistent.

Analyzing Test Scores

Teachers use box plots to understand class performance and identify students who need extra help.

If a class has median 75%, Q1 at 68%, and Q3 at 85%, about 25% of students scored below 68% and may need additional support.

Q: Why use box plots instead of just the mean and range?

A: Box plots show much more information: the median (which isn't affected by outliers), how spread out the middle 50% is, whether the data is symmetric or skewed, and which values are unusual outliers. The mean and range can be misleading when extreme values exist.

Q: What if two box plots have the same median but different box sizes?

A: The median tells you the typical value, but the box size (IQR) tells you about consistency. A smaller box means the data is more clustered around the median (more consistent), while a larger box means more variability.

Q: How do I know if my data has outliers?

A: Calculate the fences: Lower fence = Q1 - 1.5 $\times$ IQR, Upper fence = Q3 + 1.5 $\times$ IQR. Any data points outside these fences are outliers and should be plotted as individual points.