Topic 1.8: Graphical Representations of Summary Statistics
Graphical Representations of Summary Statistics
Video Follow-Along Worksheet
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Learning Objectives:
Use the five-number summary to represent quantitative data graphically.
Identify and interpret the parts of a box plot, including the box, median line, whiskers, and outlier symbols.
Describe what a box plot shows about quartiles, the middle 50% of the data, and possible outliers.
Explain how the shape of a distribution affects the relationship between the mean and the median.
Essential Knowledge:
The five-number summary is the minimum, Q1, median, Q3, and maximum.
A box plot is a graphical representation of the five-number summary for quantitative data.
The box shows the middle 50% of the data, with a line at the median and ends at the quartiles.
Whiskers extend to the most extreme values that are not outliers, and outliers are shown separately.
Box plots quickly show quartiles and possible outliers, but they hide individual values and some shape details.
In a skewed-right distribution the mean is usually greater than the median, in a skewed-left distribution the mean is usually less than the median, and in a symmetric distribution they are about equal.
Key Vocabulary
Five-Number Summary
The minimum, Q1, median, Q3, and maximum taken together.
Box Plot
A graph that represents the five-number summary and possible outliers.
Quartile
One of the four equal sections of an ordered data set.
Q1
The first quartile, or the median of the lower half of the data.
Q3
The third quartile, or the median of the upper half of the data.
Median
The middle value in an ordered distribution.
Whisker
A line from the box to the most extreme value that is not an outlier.
Outlier
A value plotted beyond the whiskers because it is unusually far from the rest of the data.
Symmetric
A distribution whose left and right sides are approximately balanced.
Skewed Right
A distribution with a longer tail on the right side.
Skewed Left
A distribution with a longer tail on the left side.
Nonresistant
A statistic, such as the mean, that can be pulled by outliers.
Video 1: Graphical Representations of Summary Statistics
[0:00 – 8:04]
1.[0:09] In this video, we're going to talk about graphical representations of summary .
2.[0:21 – 0:24] One big question is how we use the five-number summary to make a .
3.[0:27 – 0:32] The second big question asks how the of the graph influences the relationship between the and the .
Flint Water Crisis Example[0:36 – 0:50] The lesson returns to the Flint water crisis and uses lead levels from water samples collected from Flint residents from January to June 2015.
4.[0:45 – 0:48] City officials measured lead levels in water samples from Flint residents.
5.[1:13 – 1:21] When making a box plot, first set up the with the variable and choose a that fits all the values.
6.[1:22] For the Flint lead-level data, the scale needs to run from to .
7.[1:27 – 1:49] The first and last values in the five-number summary are the and the .
8.[1:56 – 2:04] The middle value, or , for the Flint data is .
9.[2:15 – 2:24] The middle of the first half of the ordered data is , and for Flint its value is .
10.[2:28 – 2:38] The median of the right-hand side is , and for Flint its value is .
Five-Number Summary and Box Plot Parts[2:42 – 4:10]
Five-number summary: minimum, Q1, median, Q3, maximum
Box: extends from Q1 to Q3
Line inside the box: median
Whiskers: extend to the most extreme non-outliers
Outliers: shown separately as dots or asterisks
11.[2:54 – 3:06] Before finishing the box plot, we check for using the method, and the Flint data have of them.
12.[3:20 – 3:25] On a box plot, outliers are shown as or sometimes an .
13.[3:30 – 3:42] The box has a lower boundary at , an upper boundary at , and the marked inside it.
14.[3:48 – 3:51] The left whisker extends to the value of .
15.[3:54 – 4:10] Because there are high outliers, the right whisker goes to the largest value that is , which is .
16.[4:16 – 4:43] A box plot splits the data into quartiles, so each section contains of the data.
17.[4:54 – 5:02] One advantage is that a box plot quickly shows the and whether there are .
18.[5:17 – 5:21] One disadvantage is that a box plot does not show every individual .
19.[5:32 – 5:38] Another disadvantage is that a box plot can hide features of the distribution's .
20. Reflection: Explain how the five-number summary appears in the box plot for the Flint lead-level data. Include at least two numerical values from the lesson and mention the box, whiskers, or outliers.
Shape and Center[6:03 – 7:56]
Skewed right: mean is usually greater than the median
Skewed left: mean is usually less than the median
Symmetric: mean and median are about equal
Outliers pull the nonresistant mean more than the resistant median
21.[6:15 – 6:22] For the Flint data, the mean of is much bigger than the median of .
22.[6:26 – 6:36] The mean is because outliers pull it up, while the median is .
23.[6:40 – 6:48] In a skewed-right distribution, the mean is the median.
24.[6:50 – 7:00] In a skewed-left distribution, the mean is the median.
25.[7:02 – 7:07] In a symmetric distribution, the mean and median are about .
26. Reflection: Explain how the shape of a distribution helps you compare mean and median. Use skewed right, skewed left, and symmetric in your answer, and connect at least one part of your explanation to the Flint box plot.
Exit Ticket
A school counselor summarized one evening's homework times for 12 students. The box plot information is: minimum 12, Q1 18, median 22, Q3 31, largest non-outlier 38, and one high outlier at 58.
(a) Describe how these values would appear on a box plot, including the box, median line, whiskers, and the outlier symbol.
(b) Explain what interval contains the middle 50% of the homework times.
(c) Predict whether the mean is greater than, less than, or about equal to the median, and justify your answer using the shape of the distribution.
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