1. Honors Statistics
Day 4
Objective: Students will be able to understand and calculate variances and standard deviations. As well as determine how to describe the spread and center of a distribution based on a graph.

2. Describing Distributions with Numbers: Center/Mean
The Mean
The Average
The Arithmetical Mean
The mean is not a resistance measure of center

3. Variance
The variance ( ) of a set of observations is the average of the squares of the deviations of the observations from their mean.
The equation is: OR

4. Standard Deviation
The standard deviation (s) is the square root of the variance.
The standard deviation measures the spread by looking at how far the observations are from their mean.

5. Properties of Standard Deviation
“s” measures spread about the mean and should be used only when the mean is chosen as the measure of center.
“s” = 0 only when there is no spread. This happens when all observations have the same value. Otherwise s > 0. As the observations become more spread out about their mean, “s” gets larger.
“s”, like the mean , is strongly influenced by extreme observations. A few outliers can make “s” very large.

6. Calculating “s” by hand
Observation
(x-bar)
Deviation
(xi – x-bar)
Squared Deviation
1792
1666
1362
1614
1460
1867
1439
Sum = 11200
Mean = 1600

7. Calculating “s” by hand
Observation
(x-bar)
Deviation
(x-sub-i – x-bar)
Squared Deviation
1792
= 192
1666
=66
1362
=-238
1614
=14
1460
=-140
1867
=267
1439
=-161
Sum = 11200
Mean = 1600
Sum = 0

8. Calculating “s” by hand
Observation
(x-bar)
Deviation
(x-sub-i – x-bar)
Squared Deviation
1792
= 192
192 * 192 = 36864
1666
=66
4356
1362
=-238
56644
1614
=14
196
1460
=-140
36864
1867
=267
71289
1439
=-161
25921
Sum = 11200
Mean = 1600
Sum = 0
Sum =

9. Calculating “s” by hand

10. Calculating “s” using TI-83

11. Calculating “s” using TI-83

12. Calculating s using TI-83

13. Linear Transformations
Adding “a” to each data point does not change the shape or spread
Adding “a” to each data point does change the center by “a”

14. Linear Transformations
Multiplying each data point by “b” does change the shape or spread by a factor of “b”.
Multiplying each data point by “b” does change the center by a factor of “b”.

15. Comparing Distributions
We have already seen the usefulness of comparing two distribution in the answering difficult questions like:
“Who was a better home run hitter, Barry Bonds or Hank Aaron?”
We used side-by-side boxplots to answer that question.

16. Reminders
If there are outliers, don’t use mean or standard deviation to compare distributions.
You can’t compare a standard deviation of one distribution to the IQR of a different distribution.
You can’t compare a mean of one distribution to the median of a different distribution.

17. Complete Practice Test 1B
Homework
Complete Practice Test 1B

18. STUDY Test 9/11(A) or 9/12(B) Mean, Median, IQR (Middle 50%)
Types of Variables (Categorical, Quantitative)
Center & Spread
Standard Deviation (High, Low, Zero)
Resistant and Nonresistant Measures
Modified Boxplots
Finding Outliers
Skewed Charts and Plots
Linear Transformations