1. Introduction to Summary Statistics
Introduction to Engineering Design
Unit 3—Measurement and Statistics
Introduction to Summary Statistics
Design and Modeling
© 2012 Project Lead The Way, Inc.

2. Introduction to Summary Statistics
The collection, evaluation, and interpretation of data
Statistical analysis of measurements can help verify the quality of a design or process.

3. Introduction to Summary Statistics
Central Tendency
“Center” of a distribution
Mean
Median
Variation
Spread of values around the center
Range
Interquartile Range
Distribution
Visual summary of values
Box Plot
Summary statistics can include values that give you different kinds of information.
[click] The mean value of a variable is one type of statistic that indicates central tendency. The mean gives an indication of the center of the data. The median of a data set is another indication of central tendency.
But often we need more details on how much a quantity can vary. [click] Variation is the spread of data. The range and interquartile range are indications of distribution.
[click] Even more detail about a variable can be shown by a distribution visual, which shows a summary of the data values such as a box plot.
We will define each of these statistics and explain how to represent a set of data using some of these statistics in the form of a box plot

4. Introduction to Summary Statistics
Box Plot—Example
Median = 21
Minimum = 3
Maximum = 44
Q1 = 12
Q3 = 32
This is an example of a box plot. It provides an overview of the data set that it represents.
A box plot indicates the “center” of the data. In a box plot the “center” is the median value of the data.
A box plot also shows the spread of the data in two ways. First, the box plot shows the minimum value and the maximum value indicating the overall spread of the data values.
And it also shows the interquartile range – the “box” part of the plot. The interquartile range includes half of the data in the set.
[click] In the next activity you are going to create a box plot to represent the distance traveled by your mini-rover. For now, let’s practice with a different data set.
You will create a box plot to represent the distance that your mini-rover traveled under balloon power.

5. Introduction to Summary Statistics
Mean Central Tendency
The mean is the sum of the values of a set of data divided by the number of values in that data set.
μ = x i N
The mean is the most frequently used measure of central tendency. It is strongly influenced by outliers, which are very much greater or very much smaller than the majority of data values, that do not seem to fit with the majority of data.

6. Introduction to Summary Statistics
Mean Central Tendency
μ = x i N
μ = mean value
xi = individual data value
x i = summation of all data values
N = # of data values in the data set

7. Introduction to Summary Statistics
Mean—Example Central Tendency
Data Set
Sum of the values = 243
Number of values = 11
μ = x i N
243 11
Mean = = =

8. Introduction to Summary Statistics
A Note About Rounding in Statistics
General Rule: Don’t round a result until the final answer
If you are writing intermediate results you may round values, but keep unrounded number in the calculator memory
Mean rounding—Round to one more decimal place than the original data
Ref: Elementary Statistics, 7th edition by Bluman (McGraw-Hill, 2009).

9. Introduction to Summary Statistics
Mean Rounding —Example
Data Set
Sum of the values = 243
Number of values = 11
Reported: Mean =
μ = x i N
243 11
22. 09
Mean = = =
Notice that the bar above the 09 indicates a repeating decimal. In this case the result of the mean calculation is with the 09 repeating.
Record the mean to one more decimal place than the original data. Because the original data set is in whole numbers, record the mean to one decimal place or 22.1.
22.1

10. Median Central Tendency
Introduction to Summary Statistics
Median Central Tendency
Measure of central tendency
Data value that occurs in the middle of a set of data that has been arranged in numerical order
Symbol for median is x pronounced “x tilde” ~
The median divides the data into two sets that contain an equal number of data values.

11. Median—Example Central Tendency
Introduction to Summary Statistics
Median—Example Central Tendency
A data set that contains an odd number of values always contains a median.
Data Set
First, arrange the data in numerical order
Choose the middle value
Once in sequential order (from least to greatest), the value that falls in the middle of the data set is the median. When there is an odd number of data values, there is a single value in the “middle.”

12. Introduction to Summary Statistics
Range Variation
Measure of data variation
Difference between the greatest and least values that occur in a set of data
Symbol for range is R
Data Set
Range = R = maximum value – minimum value
R = 44 – 3 = 41

13. Interquartile Range Variation
Introduction to Summary Statistics
Interquartile Range Variation
Measure of data variation
Measure of the middle 50% of a data set OR the difference between the third quartile (Q3) and first quartile (Q1)
Symbol for interquartile range is IQR
IQR indicates how spread out the points in a data set are from the mean. If the IQR is high, it shows that the data points are more spread out; the smaller the IQR, the more bunched up the data points are around the mean.

14. Interquartile Range—Example Variation
Introduction to Summary Statistics
Interquartile Range—Example Variation
Data Set
Step 1. Put the data set in order from least to greatest.
Step 2. Find the median.

15. Interquartile Range—Example Variation
Introduction to Summary Statistics
Interquartile Range—Example Variation
Data Set
Step 3. Draw parentheses around each of the two groups of numbers above and below the median. This will help you find Q1 and Q3.
( ) ( )

16. Interquartile Range—Example Variation
Introduction to Summary Statistics
Interquartile Range—Example Variation
Data Set
Step 4. Determine Q1 and Q3.
Q1 is the median of the lower half of the data set, and Q3 is the median of the upper half of the data set.
( ) 21 ( ) Q1 Q3

17. Interquartile Range—Example Variation
Introduction to Summary Statistics
Interquartile Range—Example Variation
Data Set
Step 5. Find the difference between Q1 and Q3 to determine the IQR.
( ) 21 ( ) Q3 Q1
Interquartile Range = IQR = Q3 – Q1
IQR = 32 – 12 = 20

18. Introduction to Summary Statistics
Box Plot Distribution
A box plot, sometimes called a box and whiskers chart, is a way to show how a data set is spread out.
A box plot shows:
Q1, Q3
Minimum number in the data set
Maximum number in the data set
Median
The plot can be called a box and whiskers chart, because it resembles a box with “whiskers.”

19. Box Plot—Example Distribution
Introduction to Summary Statistics
Box Plot—Example Distribution
Data Set:
Median = 21
Minimum = 3
Maximum = 44
The first quartile, Q1, is at the far left of the box while the third quartile, Q3, is at the far right of the box.
Minimum (the least number in the data set) is at the far left of the chart, at the end of the left “whisker.”
Maximum (the greatest number in the data set) is at the far right of the chart, at the end of the right “whisker.”
The median is shown as a vertical line in the center of the box.
Q1 = 12
Q3 = 32

20. Example 2 – even number of data points
Introduction to Summary Statistics
Example 2 – even number of data points
Let’s add a number to the data set 31
Data Set
If we add a data value to our data set, there will be twelve data values. Twelve is an even number, so once the data is arranged in numerical order, there is no “middle” number.

21. Introduction to Summary Statistics
Example 2 – Median
Find the median value
Middle of data set
Data Set:
If there is an even number of data values, the “middle” of the data falls between two values. [click]
The median is the average of the two values on either side of the “middle” point. [click]
Allow students to calculate the average of the two “middle” numbers and record the calculation on their activity sheet before showing the final calculation. [click]
Median = x = = 22

22. Introduction to Summary Statistics
Example 2 – Interquartile Range
Step 1. Put the data set in order from least to greatest.
Step 2. Mark the center of the data.
Data Set
Middle of data set
Step 3. Draw parentheses around the group of numbers above and below the middle line. This will help you find Q1 and Q3.
( ) ( )

23. Introduction to Summary Statistics
Example 2 – Interquartile Range
Data Set:
Step 4. Determine Q1 and Q3, Recall how to find the median with an even set of numbers.
( ) ( )
To find Q1 and Q3 remember that if there is an even number of data values, the “middle” of the data falls between two values. The median is the average of the two values adjacent to the “middle” point. [click]
Allow students to calculate and record their calculations on the activity sheets before showing the final calculations of Q1 and Q3. [click twice for Q1] [click twice for Q2]
Q1 = 14.5
Q3 = 31.5

24. Introduction to Summary Statistics
Example 2 – Interquartile Range
Data Set
Step 5. Find the difference between Q1 and Q3 to determine the IQR.
Interquartile Range = IQR = Q3 – Q1
Allow students time to calculate the IQR before showing the calculation. [click]
IQR = 31.5 – = 17

25. Introduction to Summary Statistics
Example 2 – Box Plot
Data Set
Median = 22
Minimum = 3
Maximum = 44
Before you click to show the completed box plot, allow students time to create their box plot.
Q1 = 14.5
Q3 = 31.5