1. Variability

2. Defining Variability A measure of the spread of data . . . SUCS(C)
General: the extent that data points differ from the pattern
Univariate: the extent that data points in a statistical distribution diverge from the mean value

3. Why is variability important?
It’s a distinct characteristic of the data we are examining, just like shape, unusual features (clusters, gaps, outliers) and center
Allows us to distinguish between usual & unusual values

4. In the real world . . . Variability = Risk Risk of error
Risk of loss or decline
Risk of quality defect
Risk of missed dead line
Risk of misdiagnosis
Risk of misunderstanding

5. Variability illustrated (Using Bivariate data)

6. Measures of Variability
Range (max - min)
IQR (Q3 - Q1)
Deviations (x - x̅)
MAD 𝟏 𝒏  |x - x̅ |
Variance VAR (σ2)
Std.Dev S.D. (σ)
Lower case Greek letter sigma

7. TO BE EXPLICIT Symmetric All Other Shapes include:
Approximately normal
t-distribution
Uniform
Skewed
Bimodal
Multimodal
Geometric
Central Tendency
Mean
Median
Measures of Spread
Std.Dev (s or )
VAR
MAD
Range (Q4 – Q0)
IQR (Q3-Q1)
A relevant calc for Spread
% of points within 1s (should be 68% if normal)
Ratio of Range to IQR (2:1 is tight; 3:1 is large; 4:1 is huge)

8. Suppose that we have these data values:
Find the mean:
Find the deviations.
What is the sum of the deviations from the mean?

9.
Square the deviations:
Find the average of the squared deviations:

10. The average of the SQUARED DEVIATIONS is called the variance.
parameter
statistic
Population Sample

11. Calculation of variance of a sampledf

12. Concept: Degrees of Freedom (df)
“n” deviations contain (n - 1) independent pieces of information about variability
We’ll revisit d.f. in regression and inference (oh, joy!!!!!!!!)

13. A standard deviation is a measure of the average deviation from the mean.

14. Calculation of standard deviation

15. Use calculator Suppose that we have these data values: 24 34 26 30 37

16. Concept: “Resistance”
From U.Cal (Berkley) “a statistics is said to be resistance if a corrupting datum cannot change the statistic much”

17. Which measure(s) are resistant?
Mean?
Nope
Range?
Nope
Median?
Yes
IQR?
Yes
Std.Dev?
Nope

18. Two last concepts today for variability: 1. Linear Transformations 2
Two last concepts today for variability: 1. Linear Transformations 2. Combining Data Sets

19. Linear transformation rule
When adding a constant to a random variable, the mean changes but not the standard deviation.
When multiplying a constant to a random variable, the mean and the standard deviation changes.

20. An appliance repair shop charges a $30 service call to go to a home for a repair. It also charges $25 per hour for labor. From past history, the average length of repairs is 1 hour 15 minutes (1.25 hours) with standard deviation of 20 minutes (1/3 hour). Including the charge for the service call, what is the mean and standard deviation for the charges for labor?

21. Rules for Combining two set variables
To find the mean for the sum (or difference), add (or subtract) the two means
To find the standard deviation of the sum (or differences), ALWAYS add the variances, then take the square root.
Formulas:
If variables are independent

22. Phase Mean SD Unpacking 3.5 0.7 Assembly 21.8 2.4 Tuning 12.3 2.7
Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the times for each setup phase are independent with the following means & standard deviations (in minutes). What are the mean and standard deviation for the total bicycle setup times?
Phase
Mean SD
Unpacking
3.5
0.7
Assembly
21.8
2.4
Tuning
12.3
2.7

23. In Summary: Variability
Defined what variability is AND six different measurements
Distinguished between parameter & statistic **
Introduced concept of “df” **
Defined “resistance”
Introduced linear transformations **
Introduced combining data sets **
** More to come!

24. For AP Testing Variability is a basic building block: ALL FORMS
Need to master combo rules. Common question is to combine data sets together.
Common Mistake: Cannot directly combine Std.Dev. Need to combine variances, then take square root