Variability

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

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

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

Variability illustrated (Using Bivariate data)

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

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)

Suppose that we have these data values: 24 34 26 30 37 16 28 21 35 29 Find the mean: Find the deviations. What is the sum of the deviations from the mean?

24 34 26 30 37 16 28 21 35 29 Square the deviations: Find the average of the squared deviations:

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

Calculation of variance of a sample df

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!!!!!!!!)

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

Calculation of standard deviation

Use calculator Suppose that we have these data values: 24 34 26 30 37 24 34 26 30 37 16 28 21 35 29

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

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

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

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.

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?

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

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

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!

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