Stats LEC 2 - DESCRIBING DISTRIBUTIONS NUMERICALLY 1...

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DESCRIBING DISTRIBUTIONS
NUMERICALLY

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Numerical Summaries
Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Range
Interquartile Range
Central Tendency Variation

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Central Tendency
Mean Median Mode
n
x
x
n
1 i
i
!
=
=
Midpoint of
ranked values
Most frequently
observed value
Arithmetic
average
Measures of center: Mode, Median, Mean

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Measures of Center: Mode
• A measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• Used for either numerical or categorical data
• There may be no mode
• There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode

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• In an ordered list, the median is the “middle”
number (50% above, 50% below)
• Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 3
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Measures of Center: Median

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Measures of Center: Median
The Median the point that divides a distribution in half: 50% of
the data on the left of the median and the other 50% - on the
right.
To find the median, arrange the data in ascending (descending) order.
• If n is odd, the median is the observation with position
• If n is even, the median is the average of the middle two values with
the positions
n 1
2
+
n n
and 1
2 2
+

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25 Toyota MR2 21
23 Mersedes-Benz SLK230 20
22 Mazda Miata 19
22 Audi TT Roadster 18
20 Mersedes-Benz SLK320 17
20 Honda S 2000 16
20 BMW Z3 Roadster 15
20 Audi TT Quattro 14
19 Porsche Boxter 13
19 BMW Z3 Coupe 12
18 Chrysler Prowler 11
18 Chevrolet Corvette 10
17 Ford Thunderbird 9
17 BMW M Coupe 8
17 Acura NSX 7
16 Mersedes-Benz SL500 6
15 Porsche 911 GT2 5
13 Mersedes-Benz SL600 4
13 BMW Z8 3
11 Ferrari 360 Modena 2
9 Lamborghini Murcielago 1
City mpg Model
50% MEDIAN
0% MIN
100% MAX
25% 1st QUARTILE
75% 3rd QUARTILE

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9 Min
16 Q1
18 Median
20 Q3
25 Max
5 - Number Summary
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2
2
5

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Boxplot is a visual representation of the
five number summary.
There are four steps to built a boxplot:
1. Interquartile range IQR= Q3 – Q1, shows the spread of the middle 50% of the data
2. Fences
• lower fence LF = Q1 – 1.5(IQR)
• upper fence UF = Q3 + 1.5(IQR)
3. Whiskers - extend from Q1 and Q3 to the smallest and largest observations within
the fences
4. outliers (if any) - extreme observations that fall outside the fences

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Boxplot: gas mileage for two-seater cars
0 5 10 15 20 25 30
Collection 1
Box Plot