Download Numerical Description

DESCRIBING DISTRIBUTION NUMERICALLY
MEASURES OF CENTER:
•
MIDRANGE = (MAX + MIN) / 2
•
MEDIAN IS THE MIDDLE VALUE WITH HALF OF THE DATA ABOVE AND HALF BELOW IT.
•
MEAN = (SUM OF DATA) / (NUMBER OF COUNTS n)
EXAMPLE:
DATA: 45, 46, 49, 35, 76, 80, 89, 94, 37, 61, 62, 64, 68, 56, 57, 57, 59, 71, 72.
SORTED DATA: 35, 37, 45, 46, 49, 56, 57, 59, 61, 62, 64, 68, 71, 72, 76, 80, 89, 94.
MIDRANGE = (94 + 35) / 2 = 64.5
MEDIAN = 61
MEAN = (35 + 37 + … + 94) / 19 = 62
NOTE: FOR SKEWED DISTRIBUTIONS THE MEDIAN IS A BETTER MEASURE OF THE CENTER THAN
THE MEAN.
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MEASURES OF THE SPREAD
•
RANGE = MAX – MIN
•
INTERQUARTILE RANGE (IQR) = Q3 – Q1
Q3 = UPPER QUARTILE
= MEDIAN OF UPPER HALF OF DATA(INCLUDE MEDIAN IF n IS ODD)
Q1 = LOWER QUARTILE
MEDIAN OF LOWER HALF OF DATA(INCLUDE MEDIAN IF n IS ODD)
•
VARIANCE (later)
•
STANDARD DEVIATION (later)
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Quartiles
EXAMPLE: (odd number of observations, 19)
Median = 61
UPPER HALF
35 37 45 46 49 56 57 57 59 [61 62 64 68 71 72 76 80 89 94]
Q3 = (71 +72) / 2 = 71.5
LOWER HALF
[35 37 45 46 49 56 57 57 59 61] 62 64 68 71 72 76 80 89 94
Q1 = (49 + 56) / 2 = 52.5
IQR = 71.5 – 52.5 = 19
Note: Include the median in the calculation of both quartiles
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Quartiles
EXAMPLE: (even number of observations, 18)
35 37 45 46 49 56 57 57 59 [60] [61 62 64 68 71 72 76 80 89 ]
60 = Median = (59+61)/2 (Average of the middle two numbers)
UPPER HALF
35 37 45 46 49 56 57 57 59 [60] [61 62 64 68 71 72 76 80 89 ]
Q3 = 71
LOWER HALF
[35 37 45 46 49 56 57 57 59 ] 62 64 68 71 72 76 80 89 94
Q1 = 49
IQR = 71 – 49 = 42
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5 – NUMBER SUMMARY:
•
THE 5-NUMBER SUMMARY OF A DISTRIBUTION REPORTS ITS MEDIAN, QUARTILES, AND
EXTREMES(MINIMUM AND MAXIMUM)
•
MAX = 94
•
Q3 = 71.5
•
MEDIAN = 61
•
Q1 = 52.5
•
MIN=35
OUTLIERS: DATA VALUES WHICH ARE BEYOND FENCES
IQR = Q3 – Q1 = 19
UPPER FENCE = Q3 + 1.5IQR = 71.5 + 1.5x19 = 100
LOWER FENCE = Q1 – 1.5IQR = 52.5 – 1.5x19 = 24
IN THE EXAMPLE CONSIDERED ABOVE, THERE ARE NO OUTLIERS.
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BOXPLOTS
WHENEVER WE HAVE A 5-NUMBER SUMMARY OF A\
(QUANTITATIVE) VARIABLE, WE CAN DISPLAY THE
INFORMATION IN A BOXPLOT.
•
THE CENTER OF A BOXPLOT IS A BOX THAT SHOWS THE MIDDLE HALF OF THE
DATA, BETWEEN THE QUARTILES.
•
THE HEIGHT OF THE BOX IS EQUAL TO THE IQR.
•
IF THE MEDIAN IS ROUGHLY CENTERED BETWEEN THE QUARTILES, THEN THE
MIDDLE HALF OF THE DATA IS ROUGHLY SYMMETRIC. IF IT IS NOT CONTERED,
THE DISTRIBUTION IS SKEWED.
•
THE MAIN USE FOR BOXPLOTS IS TO COMPARE GROUPS.
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BOXPLOTS
Boxplot of C1
100
90
80
C1
70
60
50
40
30
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Examples:
• 1. Here are costs of 10 electric smoothtop ranges
rated very good or excellent by Consumers
Reports in August 2002.
•
•
850
1000
•
•
•
•
Find the following statistics by hand:
a) mean
b) median and quartiles
c) range and IQR
900
750
1400
1250
1200
1050
1050
565
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VARIANCE = “AVERAGE” SQUARE DEVIATION FROM THE MEAN
• DEVIATION = (each data value) – mean
• VARIANCE = 4648 / (19 -1) = 258.8
• STANDARD DEVIATION = SQUARE ROOT (
VARIANCE)
= 16.1
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VARIANCE = “AVERAGE” SQUARE DEVIATION FROM THE
MEAN
• Step 1: Sort Data:
565
750
850
900
1000
1050
1050
1200
1250
1400
Mean = 1001.5
Median =1025
Q1=850
Q3=1200
Range = 835
IQR= 350
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VARIANCE = “AVERAGE” SQUARE DEVIATION FROM THE
MEAN
Computing the Variance
• DEVIATION = (each data value) – mean
• Squared Deviation= ((each data value) – mean)^2
• Sum all squared deviations
• Variance = (sum of all squared deviations)/(n-1),
where n = is the number of observations
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Variance
Example:
Data
Squared Deviations
35
54.76
37
29.16
45
6.76
46
12.96
49
43.56
Mean = 42.4
•
Variance = 147.2/4 = 36.8
•
•
Std Deviation = square root of variance
Std dev = 6.06
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Some Remarks
• If the shape is skewed, report the median and
IQR.
• Mean and median will be very differnet.
• You may want to include the mean and std
deviation, but you should point out why the
mean and the median differ.
• If the histogram is symmetric, report the mean
and the std deviation and possibly the median
and IQR.
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