graphical displays of information
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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
Cou
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
Cou
nt
1
2
3
data0 1 2 3 4 5 6 7
Data Histogram
Cou
nt
1
2
3
4
5
data0 1 2 3 4 5 6 7
Data Histogram
Cou
nt1
2
3
4
5
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
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