Graphical Displays of Information Chapter 3.1 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U.

Graphical Displays of Information

Chapter 3.1 – Tools for Analyzing Data

Mathematics of Data Management (Nelson)

MDM 4U

Histograms

contain continuous data grouped in class intervals, which will display how data is spread over a range

the width of each bar is known as the bin width

different bin widths produce different shaped distributions

bin widths should be equal and there should be at least five (5)

Histogram Example these

histograms represent the same data

however, one shows much less of the structure of the data

too many bins (bin width too small) is also a problem

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SomeData40 60 80 100 120

Data Histogram

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SomeData40 60 80 100 120

Data Histogram

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Data Histogram

Histogram Applet – Old Faithfulhttp://www.isixsigma.com/offsite.asp?A=Fr&Url

=http://www.stat.sc.edu/~west/javahtml/Histogram.html

Bin Width Calculation

the bin width is calculated by dividing the range = max – min by the number of intervals you desire (5-6)

the bins should not overlap wrong: 0-10, 10-20, 20-30, 30-40

Discrete correct: 0-10, 11-20, 21-30, 31-40

Continuous correct: 0-9.99, 10-19.99, 20-29.99, 30-39.99

Mound-shaped distribution The middle interval(s) have the greatest

frequency (i.e. the tallest bar) The bars get smaller as you move out to the

edges.

U-shaped distribution

Lowest frequency in the centre, highest towards the outside

E.g. height of a combined grade 1 and 6 class

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 Normal Distributions are

symmetric

Skewed distribution (left and right) Highest frequencies at one end Left-skewed drops off to the left E.g. the years on a handful of quarters

Exercises Define in your notes:

Frequency distribution (p. 146) Cumulative frequency (p. 146) Relative frequency (p. 146)

Try page 146 #1,2,3, 11 (use Excel or Fathom),13

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:

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

1

1514...4321k

k

Examples:

write in expanded form:

= [2(4) + 1] + [2(5) + 1] + [2(6) + 1] + [2(7) + 1] = 9 + 11 + 13 + 15 =48 note that any letter can be used for the index of

summation, though k, a, n, i, j & x are often used

7

4

)12(n

n

Example: write the following in sigma notation

3210 2

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2

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2

38

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nn

The Mean

n

x

x

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ii

1

found by dividing the sum of all the data points by the number of elements of data

Deviation the distance of a data point from the mean calculated by subtracting the mean from the

value

The Weighted Mean

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ii

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w

wxx

1

1

where xi represent the data points, wi represents the weight or the frequency

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)

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

Median

the midpoint of the data calculated by placing all the values in order 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

if there is an odd number of values, the median is the middle number 1 4 6 8 9 median = 6

Mode

Simply chosen by finding 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 Multiple modes? Uniform Modes are appropriate for discrete data or non-numerical data

shoe sizes shoe colors

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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What Method is Most Appropriate? Outliers are data points that are quite

different from the other points Outliers have the greatest effect on the mean 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, use the mode

Example 1 find the mean, median and mode

mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7 median = 3 mode = 3

which way is it skewed? Left

Survey responses 1 2 3 4

Frequency 2 8 14 3

Example 2 Find the mean, median and mode

mean = [(145x3) + (155x7) + (165x4)] / 14 = 155.7 median = 155 mode = 151-160

which way is it skewed? Mound-shaped

Height 141-150 151-160 161-170

No. of Students 3 7 4

Exercises

try page 159 #4, 5, 6, 8

Remembrance Day by the Numbers http://www42.statcan.ca/smr08/smr08_064_e

.htm

References

Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page