Topic 1.7: Summary Statistics for a Quantitative Variable
Summary Statistics for a Quantitative Variable
Video Follow-Along Worksheet
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Learning Objectives:
Calculate and interpret mean and median as measures of center for quantitative data.
Find and interpret Q1, Q3, range, IQR, and standard deviation for a quantitative distribution.
Use the 1.5 IQR rule and the 2 standard deviation rule to identify possible outliers.
Choose appropriate summary statistics based on skewness, symmetry, and resistance to outliers.
Essential Knowledge:
A statistic is a numerical summary of sample data.
The mean is the sum of all data values divided by the number of values, and the median is the middle ordered value.
Q1 and Q3 are quartiles that mark the middle 50% of the ordered data set.
Range, IQR, and standard deviation are measures of variability, and variance is the square of the standard deviation.
Mean, range, and standard deviation are nonresistant to outliers, while median and IQR are resistant.
For skewed distributions, median and IQR are preferred; for symmetric distributions, mean and standard deviation are preferred.
The average found by adding all values and dividing by the number of values.
Median
The middle value in an ordered data set.
Quartiles
Values that divide ordered data into sections, including Q1 and Q3.
Q1
The median of the lower half of the ordered data.
Q3
The median of the upper half of the ordered data.
Range
The maximum minus the minimum.
IQR
The interquartile range, found by Q3 minus Q1.
Standard Deviation
A measure of the typical distance each value is from the mean.
Variance
The square of the standard deviation.
Outlier
A value that is unusually far from the rest of the distribution.
Resistant
A statistic that is not greatly affected by outliers.
Nonresistant
A statistic that changes a lot when outliers are present or removed.
Video 1: Summary Statistics for a Quantitative Variable
[0:00 – 8:04]
1.[0:04] In this video, we are talking about summary for a quantitative variable.
2.[0:28] The primary focus is on the and the of a quantitative data set.
Flint Water Crisis Example[0:37 – 0:56] The lesson returns to the Flint water crisis data and uses lead levels from 71 water samples to practice describing center and variability with summary statistics.
3.[0:40] City officials measured lead levels in water samples from Flint residents.
4.[1:13] The is found by adding all the values and dividing by the number of values.
Measures of Center[1:13 – 2:31]
Mean: add all values and divide by the number of values.
Median: the middle ordered value.
Q1: median of the first half of the data.
Q3: median of the second half of the data.
5.[1:39] In the sample mean formula, x-bar represents the and n represents the number of .
6.[1:43 – 2:03] Another common measure of center is the , and with an even number of values you take the of the two middle values.
7.[2:31] Q1 is the median of the of the data, and Q3 is the median of the .
8.[2:59 – 3:06] For the Flint data, the values add to and the mean is parts per billion.
9.[3:32 – 3:41] With 71 ordered values, the median is the value, which is parts per billion.
10.[4:21 – 4:36] For the Flint lead levels, Q1 is and Q3 is .
Measures of Variability[4:49 – 5:40]
Statistic
Meaning
Range
Maximum minus minimum
IQR
Q3 minus Q1; spread of the middle 50%
Standard Deviation
Typical distance from the mean
Variance
Square of the standard deviation
11.[4:53 – 5:21] Three measures of variability in the lesson are the , the , and the .
12.[5:40 – 7:12] The square of the standard deviation is the . For the Flint data, the range is , the IQR is , and the standard deviation is about .
13. Reflection: Explain what the mean, median, and IQR tell you about the Flint lead-level data. Use context and include at least one numerical value from the lesson.
Video 2: Outliers, Resistant Measures, and Choosing Statistics
[0:00 – 8:30]
14.[1:39] Method 1 says a value is an outlier if it is more than IQRs below Q1 or above Q3.
15.[2:49 – 3:02] For the Flint data, any value greater than is an outlier by method 1, and there are such high outliers.
16.[3:15] Method 2 says a value is an outlier if it is two or more above or below the mean.
17.[4:00 – 4:12] Using method 2, any Flint lead level above is an outlier, so there are high outliers by this rule.
Removing the Largest Outlier[4:23 – 5:46] The teacher removes the largest Flint value, 104, and compares the old and new summary statistics to see which measures change the most.
18.[5:06 – 5:19] After removing 104, the mean drops to and the standard deviation drops to about .
19.[5:25 – 5:46] After removing 104, the median stays at and the IQR changes only to .
Resistant vs. Nonresistant[5:59 – 6:11]
Nonresistant: mean, range, standard deviation
Resistant: median, IQR
Outliers change nonresistant measures much more than resistant measures.
20.[5:59 – 6:11] The mean, range, and standard deviation are , while the median and IQR are .
21.[6:58 – 7:07] For a skewed distribution, the best measures are the for center and the for variability. For a symmetric distribution, use the and the .
22. Reflection: Explain why the median and IQR are better than the mean and standard deviation for the Flint lead-level distribution. Use the ideas of skewness, outliers, and resistant versus nonresistant measures.
Exit Ticket
A coach recorded the number of minutes 10 students waited for rides after practice.
Student
1
2
3
4
5
6
7
8
9
10
Minutes Waiting
3
4
4
5
5
5
6
6
7
22
(a) Choose the best measure of center and the best measure of variability for this distribution.
(b) Explain whether 22 minutes should be considered an outlier using one method from the lesson.
(c) State whether your chosen statistics are resistant or nonresistant, and explain why they fit this distribution.
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