Comparing two data sets helps you decide how groups are similar and different using evidence instead of guesses. In statistics, you usually compare center, spread, shape, and outliers. Comparative box plots are useful because they place the medians, quartiles, ranges, and possible outliers of two groups on the same scale.
Back-to-back stem plots are useful because they show the actual data values while making the two distributions easy to compare.
Key Facts
- Median = the middle value when the data are ordered.
- Range = maximum value - minimum value.
- IQR = Q3 - Q1.
- Mean = sum of all data values / number of data values.
- A common outlier rule is: values less than Q1 - 1.5(IQR) or greater than Q3 + 1.5(IQR) may be outliers.
- When comparing two data sets, describe center, spread, shape, and outliers using the same scale and units.
Vocabulary
- Center
- The center of a data set is a typical or middle value, often described by the median or mean.
- Spread
- Spread describes how far apart the data values are, often measured by range or interquartile range.
- Median
- The median is the middle value of an ordered data set, or the average of the two middle values if there is an even number of values.
- Interquartile range
- The interquartile range, or IQR, is the distance between the first quartile and third quartile and describes the spread of the middle half of the data.
- Outlier
- An outlier is a data value that is unusually far from the rest of the data.
Common Mistakes to Avoid
- Comparing only the highest values is wrong because the maximum does not describe the typical value or the overall distribution.
- Using different number scales for the two graphs is wrong because it can make one data set look more spread out or shifted than it really is.
- Saying the data set with the larger range is always more variable is incomplete because one extreme outlier can make the range large while most values are still close together.
- Calling the mean the best center for every data set is wrong because outliers or strong skew can pull the mean away from a typical value.
Practice Questions
- 1 Data Set A: 4, 6, 7, 8, 10, 12, 13. Data Set B: 3, 5, 5, 9, 11, 14, 16. Find the median and range of each data set, then state which set has the larger center and which has the larger spread by range.
- 2 A box plot for Group 1 has Q1 = 20, median = 28, Q3 = 34, minimum = 16, and maximum = 42. A box plot for Group 2 has Q1 = 22, median = 25, Q3 = 31, minimum = 18, and maximum = 36. Find the IQR and range for each group, then compare center and spread.
- 3 Two classes took the same quiz. Class A has a higher median score, but Class B has a much larger IQR and one very low outlier. Explain what this means about typical performance and consistency in the two classes.