1. Statistics and Probability (Day 1) 13.2 Measures of Center and Spread
Essential Question: What are the different graphical displays of data?
Statistics and Probability (Day 1) 13.2 Measures of Center and Spread

2. 13.2 Measures of Center and Spread
Mean → Average
Example 1: Mean Number of Accidents
A six-month study of a busy intersection reports the number of accidents per month as 3, 8, 5, 6, 6, 10. Find the mean number of accidents per month at the site.
Solution: Add all the values, divide by the number of values

3. 13.2 Measures of Center and Spread
Example 2, Mean Home Prices
In the real-estate section of the Sunday paper, the following houses were listed:
2-bedroom fixer-upper: $98,000
2-bedroom ranch: $136,700
3-bedroom colonial: $210,000
3-bedroom contemporary: $289,900
4-bedroom contemporary: $315,500
8-bedroom mansion: $2,456,500
Find the mean price, and discuss how well it represents the center of the data.
$584,433.33

4. 13.2 Measures of Center and Spread
Median → middle value of a data set
If the number of values is odd, the median is the number in the middle
If the number of values is even, the median is the average of the two middle numbers
Example 3: Median Home Prices
Find the median of the data set in example 2, and discuss how well it represents the center of data.

5. 13.2 Measures of Center and Spread
Example 3: Median Home Prices
Find the median of the data set in example 2, and discuss how well it represents the center of data.
2-bedroom fixer-upper: $98,000
2-bedroom ranch: $136,700
3-bedroom colonial: $210,000
3-bedroom contemporary: $289,900
4-bedroom contemporary: $315,500
8-bedroom mansion: $2,456,500
$249,950

6. 13.2 Measures of Center and Spread
Mode → data value with the highest frequency
Most often used for qualitative data
Why?
If every value appears the same number of times, there is no mode
If two or more scores have equal frequency, the data is called bimodal (2 modes), trimodal (3 modes), or multimodal.

7. 13.2 Measures of Center and Spread
Example 4: Mode of a Data Set
Find the mode of the data represented by the bar graph below

8. 13.2 Measures of Center and Spread
Mean, Median, and Mode of a Distribution
Symmetric Distribution: mean = median
Skewed Left: mean is to the left of the median
Skewed Right: mean is to the right of the median

9. 13.2 Measures of Center and Spread
Assignment
Page 862 – 863
Problems 1 – 17 (odd)

10. Statistics and Probability (Day 2) 13.2 Measures of Center and Spread
Essential Question: What are the different graphical displays of data?
Statistics and Probability (Day 2) 13.2 Measures of Center and Spread

11. 13.2 Measures of Center and Spread
Measures of Spread
Variability → spread of the data 6 5 7 8
1 9 9
1 5 10
1 5 9 11
5 9 12 13 14
most least

12. 13.2 Measures of Center and Spread
Standard Deviation: most common measure of variability
Best used with symmetric distribution (bell curve)
Measures the average distance of an element from the mean
Deviation: individual distance of an element from the mean

13. 13.2 Measures of Center and Spread
Standard Deviation
Find the mean
Determine each individual deviation
Square each individual deviation
Find the average of those squared values
This gives you the variance (σ2)
Take the square root of the variance
Denoted using the Greek letter sigma (σ)
Population versus Sample
When dealing with a sample of a population, divide by n-1 instead of n. The result is called the sample standard deviation, and is denoted by s.
As samples become larger, the deviation approaches the population standard deviation

14. 13.2 Measures of Center and Spread
Find the standard deviation for the data set:
2, 5, 7, 8, 10
Find the mean:
Find each individual deviation:
Square each individual deviation:
Find the variance:
Population? Average n:
Sample? Use n – 1:
Take square root of each:
Population standard deviation:
Sample standard deviation:
32/5 = 6.4
4.4, 1.4, 0.6, 1.6, 3.6
19.36, 1.96, 0.36, 2.56, 12.96
37.2/5 = 7.44
37.2/4 = 9.3
σ ≈ 2.73
s ≈ 3.05

15. 13.2 Measures of Center and Spread
Using the calculator
TI Calculators
Make a list (2nd, minus sign, edit)
Go into statistical functions (2nd, plus sign, Calc)
Choose “OneVar”
Go into list (2nd, minus sign, names)
Choose the appropriate list
Casio Calculators
Menu (Stat – Menu item #2)
Make a list
Calc (F2)
1Var (F1)

16. 13.2 Measures of Center and Spread
What I want you to know
What a standard deviation is
How to calculate it based on a population
How to calculate it based on a sample
What is cool (but not necessary) to know:
68% - 96% - 99% of population within standard distributions

17. 13.2 Measures of Center and Spread
Box & Whisker Plot
Need five pieces of data: minimum, Q1, median, Q3, maximum
Box is drawn, with the Q1 and Q3 representing the left and right sides of the box, respectively
Vertical line is drawn at the median
“Whiskers” are horizontal lines drawn from the left side of the box to the minimum, and right side to the maximum

18. 13.2 Measures of Center and Spread
Interquartile Range
Measure of variability that is resistant to extreme values
A median divides a data set into an upper & lower half
The first quartile, Q1, is the median of the lower half
The third quartile, Q3, is the median of the upper half
The interquartile range is the difference between the two quartiles (Q3 – Q1), which represents the spread of the middle 50% of data

19. 13.2 Measures of Center and Spread
Assignment
Page 862 – 863
Problems 19 – 37 (odd)