1
DESCRIBING DISTRIBUTIONS
NUMERICALLY
2
Numerical Summaries
Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Range
Interquartile Range
Central Tendency
Variation
3
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
4
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
5
•
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
6
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
+
7
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
8
9
Min
16
Q1
18
Median
20
Q3
25
Max
5 - Number Summary
7
2
2
5
9
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
10
Boxplot: gas mileage for two-seater cars
0
5
10
15
20
25
30
Collection 1
Box Plot
