Variability.

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Presentation transcript:

Variability

Why is the study of variability important? Distinguish between usual & unusual values We often want less variability scores on standardized tests time bombs medicine

Measures of Variability Range (max – min) Interquartile range (Q3 – Q1) Deviations Variance Standard deviation Lower case Greek letter sigma

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:

Variance: The average of the squared deviations parameter statistic Population Sample

Degrees of freedom (df) Variance of a sample Degrees of freedom (df)

Degrees of Freedom (df) n observations have n – 1 independent pieces of information about variability

Standard deviation: the average deviation from the mean

Calculation of standard deviation Use calculator!

Which measure(s) of variability is/are resistant? IQR

Activity

Linear Transformation Rule Adding a constant to all data  mean changes, but not standard deviation Multiplying a constant to all data  mean and standard deviation change

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 Variables Mean of a sum/difference  add/subtract the two means Standard deviation of a sum/difference  add variances, then square root Variables must be independent!

Bicycles arrive at a bike shop in boxes 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