1. Summary Statistics
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Summary Statistics
Last week we used stemplots and histograms to describe the shape, location, and spread of a distribution. This week we use numerical summaries of location and spread.
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2. Main Summary Statistics by Type
Central location
Mean
Median
Mode
Spread
Variance and standard deviation
Quartiles and Inter Quartile Range (IQR)
Shape
Statistical measures of spread (e.g., skewness and kurtosis) are available but are seldom used in practice (not covered)
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3. Notation n  sample size X  variable xi  value of individual i
  sum all values (capital sigma)
Illustrative example (sample.sav), data:  
n = 10
X = age
x1= 21, x2= 42, …, x10= 52
x = … + 52 = 290
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4. Sample Mean
Illustrative example: n = 10 (data & intermediate calculations on prior slide)
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5. Population Mean
Same operation as sample mean, but based on entire population (N = population size)
Not available in practice, but important conceptually
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6. Interpretation of xbar
Sample mean used to predict
an observation drawn at random from a sample
an observation drawn at random from the population
the population mean
Gravitational center (balance point)
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7. Median – a different kind of average
“Middle value”
Covered last week
Order data
Depth of median is (n+1) / 2
When n is odd  middle value
When n is even  average two middle values
Illustrative example, n = 10  median has depth (10+1) / 2 = 5.5
 median = average of 27 and 28 = 27.5
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8. Median is “robust” Robust  resistant to skews and outliers
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Median is “robust”
Robust  resistant to skews and outliers
This data set has a mean (xbar) of 1600:
This data set has an outlier and a mean of 2743:
Outlier
The median is 1614 in both instances.
The median was not influenced by the outlier.
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9. Mode Mode  value with greatest frequency
e.g., {4, 7, 7, 7, 8, 8, 9} has mode = 7
Used only in very large data sets
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10. Mean, Median, Mode Symmetrical data: mean = median
positive skew: mean > median [mean gets “pulled” by tail]
negative skew: mean < median
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11. Summary Statistics
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Spread = Variability
Variability  amount values spread above and below the average
Measures of spread
Range and inter-quartile range
Standard deviation and variance (this week)
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12. Summary Statistics
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Range = max – min
The range is rarely used in practice b/c it tends to underestimate population range and is not robust
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13. Standard deviation Deviation = Sum of squared deviations =
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Standard deviation
Most common descriptive measure of spread
Deviation =
Sum of squared deviations =
Sample variance =
Sample standard deviation =
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14. Standard deviation (formula)
Sample standard deviation s is the unbiased estimator of population standard deviation .
Population standard deviation  is rarely known in practice.
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15. Summary Statistics
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New data set (“Metabolic Rates”) This example is not in your lecture notes
Metabolic rates (cal/day), n = 7
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16. Metabolic rates showing mean (
Metabolic rates showing mean (*) and deviations of first two observations
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17. Standard Deviation Calculation metabolic.sav – introduced slide 15
Summary Statistics
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Standard Deviation Calculation metabolic.sav – introduced slide 15
Observations
Deviations
Squared deviations
1792
1792 1600 = 192
(192)2 = 36,864
1666
1666 1600 = 66
(66)2 = 4,356
1362
1362 1600 = -238
(-238)2 = 56,644
1614
1614 1600 = 14
(14)2 =
1460
1460 1600 = -140
(-140)2 = 19,600
1867
1867 1600 = 267
(267)2 = 71,289
1439
1439 1600 = -161
(-161)2 = 25,921
SUMS  0*
SS = 214,870
* Sum of deviations will always equal zero
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18. Standard Deviation Metabolic data (cont.)
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Standard Deviation Metabolic data (cont.)
Variance (s2)
Standard deviation (s)
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19. General rule for rounding means and standard deviations
Report mean to one additional decimals above that of the data
To achieve accuracy, intermediate calculations should carry still an additional decimals
Illustrative example
Suppose data is recorded with one decimal accuracy (i.e., xx.x)
Report mean with two decimal accuracy (i.e., xx.xx)
Carry all intermediate calculations with at least three decimal accuracy (i.e., xx.xxx)
Even more important: Always use common sense and judgment.
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20. TI-30XIIS – about $12
In practice, we often use software or a calculator to check our standard deviation
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21. Interpretation of Standard Deviation
Larger standard deviation  greater variability
s1 = 15 and s2 = 10  group 1 has more variability
rule – Normal data only
68% of data with 1 SD of mean, 95% within 2 SD from mean, and 99.7% within 3 SD of mean
e.g., if mean = 30 and SD = 10, then 95% of individuals are in the range 30 ± (2)(10) = 30 ± 20 = (10 to 50)
Chebychev’s rule – All data
at least 75% data within 2 SD of mean
e.g., mean = 30 and SD = 10, then at least 75% of individuals in range 30 ± (2)(10) = (10 to 50)
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22. Summary Statistics
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Quartiles and IQR
Quartiles divide the ordered data into four equally-sized groups
Q0 = minimum
Q1 = 25th %ile
Q2 = 50th %ile (Median)
Q3 = 75th %ile
Q4 = maximum
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23. gives spread of middle 50% of the data
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Rule for quartiles
Find the median  Q2
Middle of lower half of data set  Q1
Middle of upper half of the data  Q3
Bottom half | Top half |
   Q Q Q3
IQR = Q3 – Q1 = 42 – 21 = 21
gives spread of middle 50% of the data
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24. 5-Point Summary (sample.sav)
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5-Point Summary (sample.sav)
Q0 = 5 (minimum)
Q1 = 21 (lower hinge)
Q2 = 27.5 (median)
Q3 = 42 (upper hinge)
Q4 = 52 (maximum)
Best descriptive statistics for skewed data
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25. Illustrative example (metabolic.sav)
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Illustrative example (metabolic.sav)
 median
Bottom half :  Q1 = ( ) / 2 =
Top half:  Q3 = ( ) / 2 = 1729
5-point summary: 1362, , 1614, 1729, 1867
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26. Box-and-whiskers plot (boxplot)
5 point summary + “outside values”
Procedure
Determine 5-point summary
Draw box from Q1 to Q3
Draw Q2
Calculate IQR = Q3 – Q1
Calculate fences
FLower = Q1 – 1.5(IQR)
FUpper = Q (IQR)
Determine if any outside values? If so, plot separately
Determine inside values and draw whiskers from box to inside values
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27. Boxplot example 5-point: 5, 21, 27.5, 42, 52 IQR = 42 – 21 = 21
5-point: 5, 21, 27.5, 42, 52
IQR = 42 – 21 = 21
FU = 42 + (1.5)(21) = 73.5
No outside above (outside) Upper inside value = 52
FL = 21 – (1.5)(21) = –10.5
No values below (outside)
Lower inside value = 5 60 50 40 30 20 10
Upper inside = 52
Q3 = 42
Q1 = 21
Lower inside = 5
Q2 = 27.5
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28. Boxplot example 2 5-point: 3, 22, 25.5, 29, 51 IQR = 29 – 22 = 7
5-point: 3, 22, 25.5, 29, 51
IQR = 29 – 22 = 7
FU = 29 + (1.5)(7) = 39.5
One outside (51)
Inside value = 31
FL = 22 – (1.5)(7) = 11.5
One outside (3)
Inside value = 21
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29. Boxplot example 3 (metabolic.sav)
5-point: 1362, , 1614, 1729, 1867 (slide 30)
IQR = 1729 – = 279.5
FU = (1.5)(279.5) =
None outside
Upper inside = 1867
FL = – (1.5)(279.5) =
Lower inside = 1362
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30. Interpretation of boxplots
Location
Position of median
Position of box
Spread
Hinge-spread (box length) = IQR
Whisker-to-whisker spread (range or range minus the outside values)
Shape
Symmetry of box
Size of whiskers
Outside values (potential outliers)
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31. Side-by-side boxplots
Boxplots are especially useful for comparing groups:
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