Lesson 1.9 · Unit 1 · CED topic 1.9
Comparing distributions and standardized scores
Statistics is at its most useful when it compares: one class with another, treated patients with untreated ones, this year with last year. This lesson shows how to compare the distributions of one quantitative variable using graphs and numbers, and introduces the z-score, a way of describing where a single value sits in its own distribution so that values from different distributions can be compared fairly.
Learning objectives.
- 1.9.A. Compare multiple quantitative one-variable graphical representations.
- 1.9.B. Compare multiple quantitative one-variable graphical representations of summary statistics.
- 1.9.C. Justify a claim using multiple quantitative one-variable graphical representations.
- 1.9.D. Calculate z-scores with population parameters.
- 1.9.E. Compare z-scores as measures of relative position for distributions.
Essential knowledge.
- 1.9.A.1. Graphical representations of a quantitative variable can be used to compare important features between two or more distributions of the same quantitative variable. Histograms, back-to-back stem-and-leaf plots, and dotplots may be used to compare center, variability, shape, outliers, clusters, or gaps in two or more distributions. Boxplots may be used to compare center, variability, outliers, and skewness (or symmetry).
- 1.9.B.1. A comparison of graphical representations for two or more distributions can include any of the numerical summaries (e.g., mean, standard deviation, etc.).
- 1.9.C.1. Multiple quantitative one-variable graphical representations may reveal information that can be used to justify claims about the variable in context.
- 1.9.D.1. A standardized score measures the number of standard deviations a data value falls above or below the mean.
- 1.9.D.2. A z-score is calculated as \(z=\dfrac{x_i-\mu}{\sigma}\), where \(x_i\) is the data value, \(\mu\) is the population mean, and \(\sigma\) is the population standard deviation. A z-score measures how many standard deviations a data value is above (positive z-score) or below (negative z-score) the mean. When the population mean and standard deviation are unknown, the sample mean and standard deviation may be used to determine a z-score.
- 1.9.E.1. z-scores may be used to compare relative positions of individual values within a distribution or between distributions.
Comparing distributions with graphs
Graphical representations of a quantitative variable can be used to compare important features of two or more distributions of the same variable.
- Histograms, back-to-back stem-and-leaf plots, and dotplots can be used to compare center, variability, shape, outliers, clusters and gaps.
- Boxplots can be used to compare center, variability, outliers and skewness (or symmetry).
To compare, put the distributions on the same scale: stacked dotplots or parallel boxplots share one axis, and a back-to-back stem-and-leaf plot shares one set of stems. Here are the scores on the same exam for two sections of a course, 15 students in each (Section A met in person; Section B met online).
A back-to-back stem-and-leaf plot puts the leaves for one group on the left of the stems and the other group on the right.
| Section A (leaves) | Stem | Section B (leaves) |
|---|---|---|
| 4 | 8 | |
| 5 | 5 | |
| 8 2 | 6 | 0 6 |
| 9 8 6 5 4 2 0 | 7 | 0 3 5 8 |
| 8 5 3 1 0 | 8 | 0 4 8 |
| 2 | 9 | 0 4 6 |
| 10 | 0 |
Key: \(7\mid3\) on the right means a score of 73 in Section B. A boxplot makes the same comparison compactly.
Comparing numerical summaries
A comparison of graphical representations can include any numerical summary: means, medians, standard deviations, IQRs, the five-number summary. For the two sections, Section A has five-number summary 62, 72, 78, 83, 92 and Section B has 48, 66, 78, 90, 100. Both have median 78, but the IQR is 11 for Section A and 24 for Section B, and the range is 30 for A and 52 for B.
Writing a comparison
A comparison has to refer to both distributions in each sentence and use a comparison word (higher, lower, more variable, wider, similar). Listing the features of one group and then the other, without connecting them, is not a comparison. Cover shape, center, variability and unusual features, and tie each to the context.
Compare the exam scores in Section A and Section B.
Center: the typical score is about the same in the two sections: both medians are 78 points. Variability: the scores in Section B are much more spread out than in Section A (IQR 24 points compared with 11 points, and range 52 compared with 30). Shape: Section A's scores are roughly symmetric and clustered between about 70 and 85, while Section B's scores are spread fairly evenly from the high 40s to 100. Unusual features: neither section has outliers, but Section B has both the lowest (48) and the highest (100) scores. So a student in Section B is more likely to score very high or very low than a student in Section A, even though the middle score is the same.
z-scores: where does a value stand?
A standardized score measures the number of standard deviations a data value falls above or below the mean. The z-score of a data value \(x_i\) is \[z=\frac{x_i-\mu}{\sigma},\] where \(\mu\) is the population mean and \(\sigma\) is the population standard deviation. A positive z-score means the value is above the mean and a negative z-score means it is below the mean; \(|z|\) tells how many standard deviations away it is. When the population mean and standard deviation are unknown, the sample mean and sample standard deviation may be used to determine a z-score: \(z=(x_i-\bar x)/s\).
Because a z-score is a count of standard deviations, it has no units, and it can be used to compare relative positions of individual values within a distribution or between distributions.
A z-score of 0 means the value equals the mean. Values with \(|z|\) greater than 2 are unusual in many (not all) data sets, which is the basis of the 2-standard-deviation outlier rule from the last lesson.
Maya scored 1280 on a college-entrance exam with \(\mu=1050\) and \(\sigma=210\). Jonas scored 29 on a different exam with \(\mu=21\) and \(\sigma=5.5\). Who did better relative to the students who took each exam?
Maya: \(z=\dfrac{1280-1050}{210}=\dfrac{230}{210}\approx1.10\). Jonas: \(z=\dfrac{29-21}{5.5}=\dfrac{8}{5.5}\approx1.45\). Jonas's score is about 1.45 standard deviations above the mean of his exam, and Maya's is about 1.10 standard deviations above the mean of hers, so Jonas did better relative to his group, even though 29 is a smaller number than 1280. Comparing raw scores across different scales would be meaningless.
For the 30 commute times, \(\bar x\approx26.03\) and \(s\approx18.04\). Find the z-scores of the 60-minute and 90-minute commutes.
\(z_{60}=\dfrac{60-26.03}{18.04}\approx1.88\) and \(z_{90}=\dfrac{90-26.03}{18.04}\approx3.55\). The 90-minute commute is about 3.55 standard deviations above the mean, which is very unusual in this data set; the 60-minute commute is 1.88 standard deviations above the mean, which is not more than 2 and so is not an outlier by the 2-standard-deviation rule (although the 1.5 × IQR rule flags it).
Scores on a test have mean 500 and standard deviation 100. What score has a z-score of \(-1.3\)?
\(z=\dfrac{x-500}{100}=-1.3\), so \(x=500+(-1.3)(100)=370\). A score of 370 is 1.3 standard deviations below the mean.
A 17-year-old boy is 1.82 m tall; boys that age have mean height 1.75 m and standard deviation 0.07 m. A 17-year-old girl is 1.72 m tall; girls that age have mean 1.65 m and standard deviation 0.06 m. Who is taller relative to their group?
Boy: \(z=(1.82-1.75)/0.07=1.00\). Girl: \(z=(1.72-1.65)/0.06\approx1.17\). The girl is 1.17 standard deviations above the mean of the girls, and the boy is 1.00 standard deviations above the mean of the boys, so she is taller relative to her group, although the boy is taller in meters.
- To compare distributions, use the same scale; mention shape, center, variability and unusual features, and write sentences that refer to both groups with comparison words.
- Histograms, dotplots and back-to-back stemplots show shape, clusters, gaps and outliers; boxplots show center, spread, outliers and skew (not peaks).
- \(z=\dfrac{x-\mu}{\sigma}\) (or \(\dfrac{x-\bar x}{s}\) from a sample) counts standard deviations from the mean; compare z-scores to compare relative position across different distributions.
- Interpret a z-score in context: "is 1.45 standard deviations above the mean score on that exam."
- A value of 71 comes from a distribution with mean 64 and standard deviation 3.5. Find and interpret its z-score.
Show answer
\(z=(71-64)/3.5=2.0\). The value is 2 standard deviations above the mean.
- Scores have mean 72 and standard deviation 8. What score has a z-score of 1.25?
Show answer
\(x=72+1.25(8)=82\).
- Anna scored 84 on a test with mean 78 and standard deviation 4. Ben scored 91 on a different test with mean 80 and standard deviation 10. Who did better relative to their class?
Show answer
Anna: \(z=(84-78)/4=1.5\). Ben: \(z=(91-80)/10=1.1\). Anna did better relative to her class, since her score is 1.5 standard deviations above the mean and Ben's is 1.1.
- A student says that a z-score of \(-2\) means the value is 2 points below the mean. Correct the statement.
Show answer
A z-score of \(-2\) means the value is 2 standard deviations below the mean. The number of points depends on the standard deviation: for example, with \(\sigma=5\) it would be 10 points below.
- Team A's daily sales (in thousands of dollars) have five-number summary 12, 15, 20, 24, 30, and Team B's have 10, 16, 20, 32, 48. Compare the two teams.
Show answer
Both teams have the same median, 20 thousand dollars, so their typical days are alike. Team B is more variable: its IQR is 16 thousand compared with 9 thousand for Team A, and its range is 38 compared with 18. Team B's distribution is also more skewed to the right, with a longer upper whisker (32 to 48) than lower whisker (10 to 16).
- Why can a z-score be used to compare values from two distributions with different units?
Show answer
A z-score is a number of standard deviations. The units of the value and of the standard deviation cancel, so it is unitless, and it describes relative position within each group.