Graphical Displays of Information

Chapter 3.1 – Tools for Analyzing DataMathematics 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 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

Cou

nt

5

10

15

20

25

30

SomeData40 60 80 100 120

Data Histogram

Cou

nt

1

23

45

6

78

9

SomeData40 60 80 100 120

Data Histogram

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nt1

2

3

4

5

6

SomeData30 40 50 60 70 80 90 100 110

Data Histogram

Histogram Applet – Old Faithfulhttp://www.stat.sc.edu/~west/javahtml/Histogram.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

105.5-

110.5

110.5-

115.5

115.5-

120.5

120.5-

125.5

125.5-

130.5

130.5-

135.5

135.5-

140.5

140.5-

145.5

145.5-

150.5

150.5-

155.5

155.5-

160.5

160.5-

165.6

0

2

4

6

8

10

12

Student Heights

Height (cm)

Frequency

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)

1

2

3

4

5

6

7

Mark0 20 40 60 80 100

Collection 1 Histogram

Measures of Central Tendency

Chapter 3.2 – Tools for Analyzing DataMathematics 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:

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”

15

1k

k

Example 1:

write in expanded form:

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

7

4

)12(n

n

Example 2: write the following in sigma notation

83

43

233

3

0 23

nn

The Mean

n

xx

n

ii

1

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. xx

The Weighted Mean

n

ii

n

iii

w

wxx

1

1

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 1 4 6 8 9 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 12 median = 7

not affected greatly by outliers

Mode 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

symmetric (mound shaped) mean = median = mode

right skewed mean > median > mode

left skewed mean < median < mode

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2

3

data0 1 2 3 4 5 6 7

Data Histogram

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2

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5

data0 1 2 3 4 5 6 7

Data Histogram

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data0 1 2 3 4 5 6 7

Data Histogram

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

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

Survey responses 1 2 3 4Frequency 2 8 14 3

Example 4 Find the mean, median and mode

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

MSIP / Homework: p. 159 #4, 5, 6, 8, 10-13

Height 141-150 151-160 161-170No. of Students 3 7 4

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