1. Compare and contrast histograms to bar graphs
Warm-up
Compare and contrast histograms to bar graphs

2. Section 1.3 Describing Quantitative Data with Numbers
Learning Objectives
After this section, you should be able to…
MEASURE center with the mean and median
MEASURE spread with standard deviation and interquartile range
IDENTIFY outliers
CONSTRUCT a boxplot using the five-number summary
CALCULATE numerical summaries with technology

3. Chapter 1: Exploring Data
Section 1.3
Describing Quantitative Data with Numbers

4. Warm-up
The 2009 roster of the Dallas Cowboys professional football team included 7 defensive linemen. Their weights (in pounds) were 306, 305, 315, 303, 318, 309, and 285. Calculate the mean and median. Show your work. Interpret your result in context.

5. Chapter 1 Exploring Data
Introduction: Data Analysis: Making Sense of Data
1.1 Analyzing Categorical Data
1.2 Displaying Quantitative Data with Graphs
1.3 Describing Quantitative Data with Numbers

6. Section 1.3 Describing Quantitative Data with Numbers
Learning Objectives
After this section, you should be able to…
MEASURE center with the mean and median
ANALYZE center with the mean and median of class data

7. Describing Quantitative Data
Measuring Center: The Mean
The most common measure of center is the ordinary arithmetic average, or mean.
Describing Quantitative Data
Definition:
To find the mean (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x1, x2, x3, …, xn, their mean is:
In mathematics, the capital Greek letter Σis short for “add them all up.” Therefore, the formula for the mean can be written in more compact notation:

8. Describing Quantitative Data
Measuring Center: The Median
Another common measure of center is the median. In section 1.2, we learned that the median describes the midpoint of a distribution.
Describing Quantitative Data
Definition:
The median M is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger.
To find the median of a distribution:
Arrange all observations from smallest to largest.
If the number of observations n is odd, the median M is the center observation in the ordered list.
If the number of observations n is even, the median M is the average of the two center observations in the ordered list.

9. Describing Quantitative Data
Measuring Center
Use the data below to calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers.
Describing Quantitative Data 10 30 5 25 40 20 15 85 65 60 45
0 5
3 00 5 7
8 5
Key: 4|5 represents a New York worker who reported a 45- minute travel time to work.

10. Describing Quantitative Data
Comparing the Mean and the Median
The mean and median measure center in different ways, and both are useful.
Don’t confuse the “average” value of a variable (the mean) with its “typical” value, which we might describe by the median.
Describing Quantitative Data
Comparing the Mean and the Median
The mean and median of a roughly symmetric distribution are close together.
If the distribution is exactly symmetric, the mean and median are exactly the same.
In a skewed distribution, the mean is usually farther out in the long tail than is the median.

11. Section 1.3 Describing Quantitative Data with Numbers
Learning Objectives
After this section, you should be able to…
MEASURE spread with interquartile range
IDENTIFY outliers

12. Describing Quantitative Data
Measuring Spread: The Interquartile Range (IQR)
A measure of center alone can be misleading.
A useful numerical description of a distribution requires both a measure of center and a measure of spread.
Describing Quantitative Data
How to Calculate the Quartiles and the Interquartile Range
To calculate the quartiles:
Arrange the observations in increasing order and locate the median M.
The first quartile Q1 is the median of the observations located to the left of the median in the ordered list.
The third quartile Q3 is the median of the observations located to the right of the median in the ordered list.
The interquartile range (IQR) is defined as:
IQR = Q3 – Q1

13. Describing Quantitative Data
Find and Interpret the IQR
Describing Quantitative Data
Travel times to work for 20 randomly selected New Yorkers 10 30 5 25 40 20 15 85 65 60 45 5 10 15 20 25 30 40 45 60 65 85 5 10 15 20 25 30 40 45 60 65 85
Q1 = 15
M = 22.5
Q3= 42.5
IQR = Q3 – Q1
= 42.5 – 15
= 27.5 minutes
Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes.

14. Section 1.3 Describing Quantitative Data with Numbers
Summary
In this section, we learned that…
When you use the median to indicate the center of a distribution, describe its spread using the quartiles.
The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q3 – Q1.
An extreme observation is an outlier if it is smaller than Q1–(1.5xIQR) or larger than Q3+(1.5xIQR) .

15. Summary Joey’s first 14 quiz grades in a marking period were
Calculate the mean. Show your work. Interpret your result in context.
Find and interpret the interquartile range (IQR).
Determine whether there are any outliers. Show your work.

16. Warm-Up
Suppose that a Major League Baseball team’s mean yearly salary for its players is $1.2 million and that the team has 25 players on its active roster. What is the team’s total annual payroll?
If you knew only the median salary, would you be able to answer this question? Why or why not?
$30 million.
No; you would not be able to calculate the team’s annual payroll from the median, because you cannot determine the sum of all 25 salaries from the median.

17. Describing Quantitative Data
Identifying Outliers
In addition to serving as a measure of spread, the interquartile range (IQR) is used as part of a rule of thumb for identifying outliers.
Describing Quantitative Data
Definition:
The 1.5 x IQR Rule for Outliers
Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile.
In the New York travel time data, we found Q1=15 minutes, Q3=42.5 minutes, and IQR=27.5 minutes.
For these data, 1.5 x IQR = 1.5(27.5) = 41.25
Q x IQR = 15 – =
Q x IQR = = 83.75
Any travel time shorter than minutes or longer than minutes is considered an outlier.
0 5
3 00 5 7
8 5

18. Warm-Up Joey’s first 14 quiz grades in a marking period were
Calculate the mean. Show your work. Interpret your result in context.
Find and interpret the interquartile range (IQR) Note: IQR=Q3-Q1

19. Section 1.3 Describing Quantitative Data with Numbers
Learning Objectives
After this section, you should be able to…
CONSTRUCT/ANALYZE a boxplot using the five-number summary

20. The Five-Number Summary
The minimum and maximum values alone tell us little about the distribution as a whole. Likewise, the median and quartiles tell us little about the tails of a distribution.
To get a quick summary of both center and spread, combine all five numbers.
Describing Quantitative Data
Definition:
The five-number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest.
Minimum Q1 M Q3 Maximum

21. Describing Quantitative Data
Boxplots (Box-and-Whisker Plots)
The five-number summary divides the distribution roughly into quarters. This leads to a new way to display quantitative data, the boxplot.
Describing Quantitative Data
How to Make a Boxplot
Draw and label a number line that includes the range of the distribution.
Draw a central box from Q1 to Q3.
Note the median M inside the box.
Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers.

22. Describing Quantitative Data
Construct a Boxplot
Consider our NY travel times data. Construct a boxplot.
Example
Describing Quantitative Data 10 30 5 25 40 20 15 85 65 60 45 5 10 15 20 25 30 40 45 60 65 85
Min=5
Q1 = 15
M = 22.5
Q3= 42.5
Max=85
Recall, this is an outlier by the
1.5 x IQR rule

23. The 2009 roster of the Dallas Cowboys professional football team included 10 offensive linemen. Their weights (in pounds) were
Find the five-number summary for these data by hand. Show your work.
Calculate the IQR. Interpret this value in context.
Determine whether there are any outliers using the 1.5 × IQR rule.
Draw a boxplot of the data.
Example

24. The data, when ordered are Therefore, the minimum is 307 and the maximum is 353. The median is the average of the 5th and 6th observations and is The first quartile is the median of the bottom 5 observations, or the 3rd observation: 311. The third quartile is the median of the top 5 observations, or the 8th observation: 326. So the five-number summary is
The IQR is 326 − 311 = 15. This is the range of the middle half of the weights of Dallas Cowboys offensive linemen in 2009.
1.5IQR = 1.5(15) = Any outliers occur below 311 − 22.5 = or above = There are no observations below However, there is one observation above 348.5: the value 353. It is an outlier.
Answers

25. Come in tomorrow with the given problems completed
HOMEWORK

26. Warm-Up
Refer to the data on travel times to work for the sample of 15 North Carolinians.
Find the mean travel time for all 15 workers

27. Section 1.3 Describing Quantitative Data with Numbers
Learning Objectives
After this section, you should be able to…
MEASURE/IDENTIFY spread with standard deviation
Check for mastery in graphing/plotting, individual, variable, mean and median

28. Homework
Correct your neighbors using pen. Make sure to make changes as needed.

29. Describing Quantitative Data
Measuring Spread: The Standard Deviation
The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Let’s explore it!
Consider the following data on the number of pets owned by a group of 9 children.
Describing Quantitative Data
Calculate the mean.
Calculate each deviation.
deviation = observation – mean
deviation: = -4
deviation: = 3
= 5

30. Describing Quantitative Data
Measuring Spread: The Standard Deviation
Describing Quantitative Data xi
(xi-mean)
(xi-mean)2 1
1 - 5 = -4
(-4)2 = 16 3
3 - 5 = -2
(-2)2 = 4 4
4 - 5 = -1
(-1)2 = 1 5
5 - 5 = 0
(0)2 = 0 7
7 - 5 = 2
(2)2 = 4 8
8 - 5 = 3
(3)2 = 9 9
9 - 5 = 4
(4)2 = 16
Sum=?
3) Square each deviation.
4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance.
5) Calculate the square root of the variance…this is the standard deviation.
“average” squared deviation = 52/(9-1) = This is the variance.
Standard deviation = square root of variance =

31. Describing Quantitative Data
Measuring Spread: The Standard Deviation
Describing Quantitative Data
Definition:
The standard deviation sx measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. This average squared distance is called the variance.

32. Describing Quantitative Data
Choosing Measures of Center and Spread
We now have a choice between two descriptions for center and spread
Mean and Standard Deviation
Median and Interquartile Range
Describing Quantitative Data
Choosing Measures of Center and Spread
The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers.
Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers.
NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA!

33. Section 1.3 Describing Quantitative Data with Numbers
Summary
In this section, we learned that…
A numerical summary of a distribution should report at least its center and spread.
The mean and median describe the center of a distribution in different ways. The mean is the average and the median is the midpoint of the values.
When you use the median to indicate the center of a distribution, describe its spread using the quartiles.
The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q3 – Q1.

34. Section 1.3 Describing Quantitative Data with Numbers
Summary
In this section, we learned that…
An extreme observation is an outlier if it is smaller than Q1–(1.5xIQR) or larger than Q3+(1.5xIQR) .
The five-number summary (min, Q1, M, Q3, max) provides a quick overall description of distribution and can be pictured using a boxplot.
The variance and its square root, the standard deviation are common measures of spread about the mean as center.
The mean and standard deviation are good descriptions for symmetric distributions without outliers. The median and IQR are a better description for skewed distributions.

35. Looking Ahead… In the next Chapter…
We’ll learn how to model distributions of data…
Describing Location in a Distribution
Normal Distributions
In the next Chapter…

36. Homework
Have the review completed when you come into class tomorrow!