1. 1.3 Describing Quantitative Data with Numbers

2. Section 1.3 After this section, you should be able to:
Measure center with 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. How long do people spend travelling to work?
The answer depends on where they live. Below are the travel times in minutes for 15 workers in North Carolina chosen by random by the Census Bureau: 30 20 10 40 25 60 15 5 12 5 1
000025 2
005 3 00 4 6
What is the mean?
What is the median?
Mean = 22.5
Median = 20

4. Measuring Center: The Mean
To find the mean, 𝑥 (pronouned x-bar) of the set of observations:
Add their values
Divide by the number of observations
Suppose we have n observations that are 𝑥 1 , 𝑥 2 , 𝑥 3 , …, 𝑥 𝑛 , their mean is:
𝑥 = 𝑠𝑢𝑚 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑛 = 𝑥 1 + 𝑥 2 , +,+…+ 𝑥 𝑛 𝑛
Compact Notation: 𝑥 = 𝑥 𝑛 𝑛

5. Measuring Center: The Median
The median M is the midpoint of the distribution
To find the median of the distribution:
Arrange all observations from smallest to largest
If the number of observations is n odd, the median M is the center observation in the ordered list.
If the number of observations is n even, the median M is the average of the two center observations in the ordered list.

6. Comparing the Mean and the Median
The mean and median measure center in different ways, and both are useful
Mean: “average” value Median: “typical” value
Relationship between mean and 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.

7. Example:
Use the data below to calculate the mean and median of commuting time (in minutes) of 20 randomly selected New York workers. 10 30 5 25 40 20 15 85 65 60 45 5 1
005555 2
0005 3 00 4
005 6 7 8
Mean = 31.25
Median = 22.5
𝑥 = … = =31.25

8. Measuring Spread: Range and Interquartile Range (IQR)
Measures variability
Single number that represents distance between maximum and the minimum values
Max – Min = range
“The data values range from 5 mins to 85 mins.” – NO, do no write this!
Instead, “The data values vary from 5 mins to 85 mins and the range is 80 mins.”

9. Interquartile Range (IQR)

10. Interquartile Range (IQR)
How to calculate quartiles:
Arrange the observations in increasing order and locate median M.
The first quartile 𝑸 𝟏 , is the median of the observations located to the left of median in ordered list.
The third quartile 𝑸 𝟑 , is the median of the observations located to the right of median in ordered list.
The interquartile range (IQR) is defined as:
𝐼𝑄𝑅= 𝑄 3 − 𝑄 1

11. Identifying Outliers 1.5 x IQR Rule for Outliers
Interquartile range is a rule of thumb for identifying outliers.
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.

12. North Carolina Workers Example
𝑄 1
𝑄 3
Median
𝐼𝑄𝑅=𝑄 3 − 𝑄 1 =30−10=20
1.5 ×𝐼𝑄𝑅= =30
Any value greater than this is an outlier
𝑄 (𝐼𝑄𝑅)=30+30=60
Any value less than this is an outlier.
𝑄 1 −1.5(𝐼𝑄𝑅)=10−30=−20