Lesson 2: Measures of Center, Percentiles and Boxplots

August 27, 2018

Review:

• Math Assessment Results
  • mean = 7.8
  • median = 8
  • mode = 8
  • standard deviation = 2.2
• Stemplots (JH: put height data on whiteboard)
• Histograms
• Attendance

Presentation:

• Measures of Center
  • Mean
    • Same as Average
    • Sum of values divided by count, or ∑x/n
    • Example with height data
  • Median
    • Center value within ordered sequence of values
    • Same as 50th Percentile
    • Position = (n + 1)/2
      • n = total number of observations
      • Another approach with larger data sets, position = 0.50*n
      • If the position is between two numbers then take the average of those two values.
    • Example with height data
  • Video: Measures of Center (4:40-7:55)

• Percentiles and Boxplots
  • Percentiles
  • To find the xth percentile
    • calculate x/100 * n (rounded to the nearest integer or take average of two values on either side)
    • result is the position of the value in an ordered (smallest to largest) data set
    • 25th percentile = 25/100 * n
    • 50th percentile = 50/100 * n = Median
    • 75th percentile = 75/100 * n
    • Example with Female height data
      • 152, 157, 160, 165(2), 168, 170(3), 173(2), 180
  • Boxplots
  • Five number summary
    • 25th Percentile = Q1
    • 50th Percentile = Q2 = Median
    • 75th Percentile = Q3
    • Minimum value
    • Maximum value
    • Range = Max – Min
    • Interquartile Range (IQR) = Q3 – Q1
    • Example with Male height data
      • 160, 168, 170(2), 173(3), 175(3), 177, 178(15), 180(4), 183(5), 185(4), 188(5), 191(4), 193, 196(2), 203
  • Video: Boxplots (3:45-8:15)
  • Examples
    • Stacking Bar Charts, Histograms and Boxplots

• Google Sheets
  • Install Statistics “Add-On”
  • Demonstrate Boxplots and Percentile calculations
  • Use Student Height Data

Activity:

Problem 1.

Here are the starting salaries, in thousands of dollars, offered to 20 students who earned bachelor’s degrees in computer science in 2011.
63 56 66 77 50 53 78 55 90 65 64 69 59 76 48 54 49 68 51 50
a. Make a stemplot.
b. Find the median, mean and mode.
c. Find the five-number summary.
d. Make a boxplot.
e. Compute the range and interquartile range (IQR).

Problem 2.

A consumer testing lab measured calories per hot dog in 20 brands of beef hot dogs. Here are the results:
186 181 176 149 184 190 158 139 175 148 152 111 141 153 190 157 131 149 135 132
a. Make a stemplot.
b. Find the median, mean and mode.
c. Find the five-number summary.
d. Make a boxplot.
e. Compute the range and interquartile range (IQR).

These problem descriptions (#1 and #2) are from “Against All Odds”, modified slightly and copied here for convenience.

Problem 3.

Use the MPG data for top selling midsize cars in the US and European markets from Lesson 1.

a. Make a back-to-back stemplot.
b. Find the five-number summaries.
c. Produce 2 boxplots, one for each market, and put them both on the same axes to facilitate comparison.

Assignment:

• Boxplots in Google Sheets
  • Use the data from Problems 1-3 (above) to make corresponding boxplots
  • You will need to install the “Statistics” Add-On
• Text: Ch. 1 p. 30-38