1. Measures of Dispersion How “spread-out” are the data?
To accompany Hawkes lesson 3.2a
Original content by D.R.S.
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2. Range = Highest - Lowest
Compute the Range of the Fernwood home sales: 3 common data.pptx Slide #2
What makes the Range change? Experiment with data in 3 common data.xlsx on the “Fernwood experiment” sheet.
Good: Easy to calculate and to understand
Bad: Sensitive to presence of abnormal values
Bad: Insensitive to the middle
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3. Variance 𝜎 2 = 𝑥 𝑖 − 𝑥 2 𝑛 or 𝑠 2 = 𝑥 𝑖 − 𝑥 2 𝑛−1
Note you have two different formulas!
Use 𝜎 2 if you’re computing for a POPULATION.
Use 𝑠 2 if you’re working for a mere SAMPLE.
The problem should make it clear as to which one should be used.
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4. Variance 𝜎 2 = 𝑥 𝑖 − 𝑥 2 𝑛 or 𝑠 2 = 𝑥 𝑖 − 𝑥 2 𝑛−1
Similar to two different names for the mean:
Greek 𝜇 for a population mean.
Roman 𝑥 for a sample mean.
But again, with Variance we have two different formulas. (With mean, the formulas were the same.)
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5. Standard Deviation 𝜎= 𝜎 2 or 𝑠= 𝑠 2 Calculate the Variance
Either 𝜎 2 for a population or 𝑠 2 for a sample
Then take the Square Root of the Variance
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6. Variance – How the formula works
Same numerator in each formula.
( 𝑥 𝑖 − 𝑥 ) measures how far away from the mean is each individual value?
The makes everything positive.
Near the mean – smaller contribution
Far away from the mean – larger contribution
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7. Variance 𝜎 2 = 𝑥 𝑖 − 𝑥 2 𝑛 or 𝑠 2 = 𝑥 𝑖 − 𝑥 2 𝑛−1
What difference does it make when you divide by 𝑛−1 instead of dividing by 𝑛 ?
So the population variance and standard deviation will always be ______er the sample’s variance and standard deviation.
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8. A Tale of Two Variances Two data sets, both with mean = 100
Values closer to the mean
Values farther from the mean
70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130
Find the sample standard deviation: 𝑠= ________
Find the sample variance: 𝑠 2 = __________
40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160
Find the sample standard deviation: 𝑠= ________
Find the sample variance: 𝑠 2 = __________
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9. Going to Extremes Again, a data set with 𝑛=13; 𝑥 =100
But the data are really extremely spread out:
0, 0, 0, 0, 0, 0, 100, 200, 200, 200, 200, 200, 200 (0 six times, 100, then 200 six times)
Find the sample standard deviation: 𝑠= ________
Find the sample variance: 𝑠 2 = __________
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10. Fernwood example
How to compute the variance and the standard deviation for the Fernwood home data, using the formula.
Data: Slide #2 in 3 common data.pptx
Computation: Sheet “Fernwood stdev #1” in 3 common data.xlsx
Experimentation: Sheet “Fernwood stdev experiment” in 3 common data.xlsx
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11. An old variance formula
You’ll see references to this version of the formula for the sample standard deviation: 𝑠 2 = 𝑛 𝑥 2 − 𝑥 𝑛 𝑛−1
Advantage: By-hand calculations can be easier and/or more precise if one uses this formula instead of the “definition” formula.
But we will let TI-84 do all the work for us.
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12. Standard Deviation with TI-84
Put data into a TI-84 List
1-Var Stats L1
It tells you both population σx and sample Sx because the TI-84 can’t do everything for you.
You still need to know whether you’re computing for a population or a sample and choose the appropriate one.
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13. Fernwood homes with TI-84
Notice σx, the population standard deviation
Notice Sx, the sample standard deviation
If Range is needed, compute maxX – minX
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14. TI-84 doesn’t tell you the Variance
If you need the variance, take the standard deviation and square it yourself.
Timesaver: VARS, 5:Statistics, 4:σx or 3:Sx
And the x2 button for one-button squaring.
Most problems are one or the other, not both.
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15. More resources (repeated from 3.1)
Link to some general summary notes about Central Tendency, Variation, and Position. (Quartiles discussion disagrees with Hawkes!)
Link to some notes about Excel functions for Central Tendency, Variation, and Position
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