Download Displaying Data Visually

Graphical Displays of
Information
Chapter 3.1 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U
Histograms

Show:






Continuous data grouped in class intervals
How data is spread over a range
Bin width = width of each bar
Different bin widths produce different shaped
distributions
Bin widths should be equal
Usually 5-6 bins
Histogram Example
Histogram
Data
9
8
7

Count
6
5
4
3
2
Histogram
Data
1
30
40
25
60
80
SomeData
100
20
15
10
5
40
60
80
SomeData
100
120
Data
Histogram
6
5
Count

These
histograms
represent the
same data
One shows
much less of
the structure
of the data
Too many
bins (bin width
too small) is
also a
problem
Count

4
3
2
1
30
40
50
60
70
80
SomeData
90
100
110
120
Histogram Applet – Old Faithful
http://www.stat.sc.edu/~west/javahtml/Histogra
m.html
Bin Width Calculation




Bin width = (range) ÷ (number of intervals)
 where range = (max) – (min)
 Number of intervals is usually 5-6
Bins should not overlap
 wrong: 0-10, 10-20, 20-30, 30-40, etc.
Discrete
 correct: 0-10, 11-20, 21-30, 31-40, etc.
 correct: 0-10.5, 10.5-20.5, 20.5-30.5, etc.
Continuous
 correct: 0-9.9, 10-19.9, 20-29.9, 30-39.9, etc.
 correct: 0-9.99, 10-19.99, 20-29.99, 30-39.99, etc.
Mound-shaped distribution



The middle interval(s) have the greatest
frequency (i.e. the tallest bars)
The bars get shorter as you move out to the
edges.
E.g. roll 2 dice
75 times
U-shaped distribution


Lowest frequency in the centre, higher towards
the outside
E.g. height of a combined grade 1 and 6 class
Student Heights
12
10
8
Frequency
6
4
2
0
Height (cm)
Uniform distribution


All bars are approximately the same height
e.g. roll a die 50 times
Symmetric distribution


A distribution that is the same on either side of the
centre
U-Shaped, Uniform and Mound-shaped
Distributions are symmetric
Skewed distribution (left or right)



Highest frequencies at one end
Left-skewed drops off to the left
E.g. the years on a handful of quarters
MSIP / Homework

Define in your notes:




Frequency distribution (p. 142-143)
Cumulative frequency (p. 148)
Relative frequency (p. 148)
Complete p. 146 #1, 2, 4 , 9, 11 (data in
Excel file on wiki),13
Warm up - Class marks

What shape is this distribution?
Which of the following can you tell from the
graph: mean? median? mode?

Left-skewed





Mean < median < mode
Modal interval: 76
(Median: 70)
(Mean: 66)
Measures of Central
Tendency
Chapter 3.2 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U
Sigma Notation



the sigma notation is used to compactly
express a mathematical series
ex: 1 + 2 + 3 + 4 + … + 15
this can be expressed: 15
k
k 1



the variable k is called the index of
summation.
the number 1 is the lower limit and the
number 15 is the upper limit
we would say: “the sum of k for k = 1 to
k = 15”
Example 1:
7

write in expanded form:
 (2n  1)
n4





This is the sum of the term 2n+1 as n takes on the
values from 4 to 7.
= (2×4 + 1) + (2×5 + 1) + (2×6 + 1) + (2×7 + 1)
= 9 + 11 + 13 + 15
= 48
NOTE: any letter can be used for the index of
summation, though a, n, i, j, k & x are the most
common
Example 2: write the following in sigma
notation
3 3 3
3  
2 4 8
3
3
 n
n 0 2
n
The Mean
x



x
i 1
i
n
Found by dividing the sum of all the data points by the
number of elements of data
Affected greatly by outliers
Deviation
 the distance of a data point from the mean
 calculated by subtracting the mean from the value
 i.e. x  x
The Weighted Mean
n
x
xw
i 1
n




i
w
i 1

i
i
where xi represent the data points, wi represents the
weight or the frequency
“The sum of the products of each item and its weight
divided by the sum of the weights”
see examples on page 153 and 154
example: 7 students have a mark of 70 and 10 students
have a mark of 80
mean = (70×7 + 80×10) ÷ (7+10) = 75.9
Means with grouped data


for data that is already grouped into class
intervals (assuming you do not have the
original data), you must use the midpoint of
each class to estimate the weighted mean
see the example on page 154-5 and today’s
Example 4
Median



the midpoint of the data
calculated by placing all the values in order
if there is an odd number of values, the median is
the middle number


median = 6
if there are an even number of values, the median is
the mean of the middle two numbers


1 4 6 8 9
1 4 6 8 9 12 median = 7
not affected greatly by outliers
Mode






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The number that occurs most often
There may be no mode, one mode, two modes (bimodal), etc.
Which distributions from yesterday have one mode?
Mound-shaped, Left/Right-Skewed
Two modes?
U-Shaped, some Symmetric
Modes are appropriate for discrete data or non-numerical data
 Eye colour
 Favourite Subject
Distributions and Central Tendancy

the relationship between the three measures
changes depending on the spread of the data
Histogram
Data
symmetric (mound shaped)

Count

3
mean = median = mode
2
1
Histogram
Data
0
5
right skewed

mean > median > mode
2
3
4
data
5
6
7
4
Count

1
3
2
1
Histogram
Data
0
1
2
3
4
data
5
6
7
5

left skewed

mean < median < mode
Count
4
3
2
1
0
1
2
3
4
data
5
6
7
What Method is Most Appropriate?






Outliers are data points that are quite
different from the other points
Outliers affect the mean the greatest
The median is least affected by outliers
Skewed data is best represented by the
median
If symmetric either median or mean
If not numeric or if the frequency is the most
critical measure, use the mode
Example 3

a) Find the mean, median and mode
Survey responses
1
2
3
4
Frequency
2
8
14
3





mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7
median = 3 (27 data points, so #14 falls in bin 3)
mode = 3
b) What shape does it have?
Left-skewed
Example 4

Find the mean, median and mode
Height
No. of Students
141-150
151-160
161-170
3
7
4

mean = [(145.5×3) + (155.5×7) + (165.5×4)] ÷ 14
= 156.2
median = 151-160 or 155.5
mode = 151-160 or 155.5
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MSIP / Homework: p. 159 #4, 5, 6, 8, 10-13


MSIP / Homework

p. 159 #4, 5, 6, 8, 10-13
References

Wikipedia (2004). Online Encyclopedia.
Retrieved September 1, 2004 from
http://en.wikipedia.org/wiki/Main_Page