Topic 1.6: Describing the Distribution of a Quantitative Variable

Learning Objectives:

Describe a quantitative distribution using shape, center, variability, and unusual features.
Use shape vocabulary such as symmetric, skewed left, skewed right, unimodal, bimodal, and uniform.
Identify outliers, gaps, and clusters in a quantitative distribution.
Describe a distribution in context rather than using disconnected graph vocabulary.

Essential Knowledge:

Descriptions of quantitative distributions should include shape, center, variability (spread), and unusual features.
Outliers are unusually small or unusually large values relative to the rest of the data.
A distribution is skewed right when the right tail is longer, skewed left when the left tail is longer, and symmetric when both sides are mirror images.
A distribution with one main peak is unimodal, with two prominent peaks is bimodal, and with about the same frequency across values is approximately uniform.
A gap is a region with no observed data values, and clusters are concentrations of values usually separated by gaps.
Descriptive statistics summarize the data set at hand and do not automatically describe a larger population.

Key Vocabulary

Distribution	The overall pattern of values in a quantitative data set.
Shape	The general form of a distribution, such as skewed or symmetric.
Center	A value that describes the typical response in a distribution.
Variability / Spread	How closely packed or spread out the data values are.
Symmetric	A distribution whose left and right sides are roughly mirror images.
Skewed Left	A distribution with a longer tail on the left side.
Skewed Right	A distribution with a longer tail on the right side.
Unimodal	A distribution with one main peak.
Bimodal	A distribution with two prominent peaks.
Uniform	A distribution where frequencies are about the same across values.
Outlier	A value that clearly does not fit with the rest of the data.
Gap	A region in a distribution where no values appear.
Cluster	A concentration of values in one part of a distribution.
Context	The real-world variable being described, which should be named in the answer.

Video 1: Describing the Distribution of a Quantitative Variable

[0:00 – 6:28]

1. [0:05] In this video, we are talking about describing the of a quantitative variable.

2. [0:15] One big question is what important we should discuss when describing quantitative data.

Flint Water Crisis Example [0:33 – 1:02] The lesson returns to the Flint water crisis data and uses a dotplot of lead levels from water samples to practice describing a quantitative distribution.

3. [0:33] The example in this lesson comes from the data.

4. [0:38] City officials measured lead levels in water samples from Flint residents.

5. [1:15] The four important characteristics are , , , and unusual .

Four-Part Description [1:15 – 1:33]

Shape tells the overall form of the graph.
Center identifies a typical value.
Variability or spread tells how tightly packed or spread out the values are.
Unusual features include outliers, gaps, and clusters.

6. [1:50] A distribution is when the left side is essentially a mirror image of the right side.

7. [2:10] A skewed left distribution has a longer tail going out to the .

8. [2:20] A skewed right distribution has the side of the distribution more drawn out.

9. [2:34] A graph with one main peak is called .

10. [2:40] A graph with two prominent peaks is called .

11. [2:43] A distribution with about the same frequency at each value is called .

Shape Vocabulary [1:43 – 2:58] Shape words describe different ideas. Skewed and symmetric describe the balance of the tails, while unimodal, bimodal, and uniform describe peaks or how evenly the values are spread across the graph.

12. [3:04] When discussing center, we ask which value best describes the .

13. [3:12] Variability asks whether values are packed close together or more .

14. [3:38] One unusual feature to watch for is , values that clearly do not fit with the rest.

15. [3:51] Other unusual features include and .

Unusual Features [3:38 – 4:07] A gap is a place where no values appear. Clusters are concentrations of values, and outliers are values that are much higher or lower than the rest of the distribution.

16. [4:18] The Flint lead-level distribution is because it has a single peak at zero.

17. [4:24] The Flint distribution is also skewed to the .

18. [4:43] A typical value for the Flint lead levels looks to be about parts per billion.

19. [4:43 – 4:55] The lead levels range from a minimum of to a maximum of parts per billion.

20. [5:00] There is a clear cluster of values between .

21. [5:06] There is a gap in the values between and .

22. [5:08] Some of the very high lead levels might be considered .

23. [5:25] When describing a distribution, you should always include the of the problem.

Putting It All Together [4:18 – 5:27]

Characteristic	Flint Example
Shape	Unimodal and skewed right
Center	Typical value around 3 to 4 parts per billion
Variability	From 0 to 104 parts per billion
Unusual Features	Cluster from 0 to 10, gap from 42 to 104, and possible outliers on the high end

24. Reflection: Describe the Flint lead-level distribution using all four characteristics from the lesson. Be sure to use context in your answer.

25. Reflection: Explain what unusual features you should look for when describing a quantitative distribution. Use at least two terms from the lesson and connect one of them to the Flint data.

Exit Ticket

A school recorded the number of minutes 11 students waited for a late bus after school.

Student	1	2	3	4	5	6	7	8	9	10	11
Minutes Waiting	1	2	3	3	4	4	4	5	5	6	18

(a) Describe the shape of the distribution using at least two appropriate terms.

(b) Give a typical value for the center in context.

(c) Describe the variability or spread in context.

(d) Identify any unusual features, such as gaps, clusters, or outliers.

Topic 1.6: Describing the Distribution of a Quantitative Variable

Key Vocabulary

Video 1: Describing the Distribution of a Quantitative Variable

Exit Ticket

Class Responses (Esc to close)