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Measures of Spread
Chapter 3.3 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U
Foot size of gongshowhockey.com users What shape are the distributions?
FreqApr1
0
10
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30
40
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8 9 10 11 12 13 13+
FreqSep5
050
100150
200250
300350
400450
8 9 10 11 12 13 13+
What is spread?
spread tells you how widely the data are dispersed
for example, the two histograms have identical mean and median, but the spread is significantly different
Co
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data2 3 4 5 6 7 8 9
data Histogram
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data Histogram
Why worry about spread? spread indicates how close the values cluster
around the middle value less spread means you have greater confidence
that values will fall within a particular range.
Vocabulary spread and dispersion refer to the same
thing range is the difference between the largest
and smallest values a quartile is one of three numerical values
that divide a group of numbers into 4 equal parts
the Interquartile Range (IQR) is the difference between the first and third quartiles
Quartiles Example
26 28 34 36 38 38 40 41 41 44 45 46 51 54 55 range = 55 – 26 = 29 Q2 = 41 Median Q1 = 36 Median of lower half of data Q3 = 46 Median of upper half of data IQR = Q3 – Q1 = 46 – 36 = 10 (contains 50% of data) if a quartile occurs between 2 values, it is
calculated as the average of the two values
A More Useful Measure of Spread The interquartile range is a somewhat useful
measure of spread Standard deviation is more useful To calculate it we need to find the mean and
the deviation for each data point Mean is easy, as we have done that before Deviation is the difference between a
particular point and the mean
Deviation
The mean of these numbers is 48 The deviation for 24 is 24 - 48 = -24 -24
12 24 36 48 60 72 84
36 The deviation for 84 is 84 - 48 = 36
Standard Deviation deviation is the distance from the piece of
data you are examining to the mean variance is a measure of spread found by
averaging the squares of the deviation calculated for each piece of data
Taking the square root of variance, you get standard deviation
Standard deviation is a very important and useful measure of spread
Standard Deviation σ² (lower case sigma
squared) is used to represent variance
σ is used to represent standard deviation
σ is commonly used to measure the spread of data, with larger values of σ indicating greater spread
we are using a population standard deviation
n
xxi
2
Example of Standard Deviation 26 28 34 36 mean = (26 + 28 + 34 + 36) / 4 = 31 σ² = (26–31)² + (28-31)² + (34-31)² + (36-31)² 4 σ² = 17 σ = √17 = 4.12
Standard Deviation with Grouped Data
grouped mean = (2×2 + 3×6 + 4×6 + 5×2) / 16 = 3.5 deviations:
2: 2 – 3.5 = -1.5 3: 3 – 3.5 = -0.5 4: 4 – 3.5 = 0.5 5: 5 – 3.5 = 1.5
σ² = 2(-1.5)² + 6(-0.5)² + 6(0.5)² + 2(1.5)² 16 σ² = 0.7499 σ = √0.7499 = 0.87
Hours of TV 2 3 4 5
Frequency 2 6 6 2
n
xxf ii
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MSIP / Homework read through the examples on pages 164-167 Complete p. 168 #2b, 3b, 4, 6, 7, 10 you are responsible for knowing how to do
simple examples by hand (<10 pieces of data)
however, we will use technology (Fathom) to calculate larger examples
have a look at your calculator and see if you have this feature (Σσn and Σσn-1)
Normal Distribution
Chapter 3.4 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U
Histograms
Histograms may be skewed...
Right-skewed Left-skewed
Histograms
... or symmetricalC
ou
nt
1
2
3
4
5
a3 4 5 6 7 8 9 10 11
Collection 1 Histogram
Normal? A normal distribution creates a histogram that is
symmetrical and has a bell shape, and is used quite a bit in statistical analyses
Also called a Gaussian Distribution It is symmetrical with equal mean, median and mode
that fall on the line of symmetry of the curve
A Real Example the heights of 600 randomly chosen Canadian
students from the “Census at School” data set the data approximates a normal distribution
0.005
0.010
0.015
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0.030
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De
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100 120 140 160 180 200 220 240Heightcm
Density = x mean s normalDensity
600 Student Heights Histogram
The 68-95-99.7% Rule area under curve is 1 (i.e. it represents 100%
of the population surveyed) approx 68% of the data falls within 1 standard
deviation of the mean approx 95% of the data falls within 2 standard
deviations of the mean approx 99.7% of the data falls within 3
standard deviations of the mean http://davidmlane.com/hyperstat/A25329.html
Distribution of Data
34% 34%
13.5% 13.5%
2.35% 2.35%
68%
95%
99.7%
x x + 1σ x + 2σ x + 3σx - 1σx - 2σx - 3σ
),(~ 2xNX
0.15%0.15%
Normal Distribution Notation
The notation above is used to describe the Normal distribution where x is the mean and σ² is the variance (square of the standard deviation)
e.g. X~N (70,82) describes a Normal distribution with mean 70 and standard deviation 8 (our class at midterm?)
),(~ 2xNX
Percentage of data between two values The area under any normal curve is 1 The percent of data that lies between two
values in a normal distribution is equivalent to the area under the normal curve between these values
See examples 2 and 3 on page 175
Why is the Normal distribution so important? Many psychological and educational
variables are distributed approximately normally: reading ability, memory, etc.
Normal distributions are statistically easy to work with All kinds of statistical tests are based on it
Lane (2003)
An example Suppose the time before burnout for an LED
averages 120 months with a standard deviation of 10 months and is approximately Normally distributed. What is the length of time a user might expect an LED to last?
95% of the data will be within 2 standard deviations of the mean
This will mean that 95% of the bulbs will be between 120 – 2×10 months and 120 + 2×10
So 95% of the bulbs will last 100-140 months
Example continued… Suppose you wanted to know how long
99.7% of the bulbs will last? This is the area covering 3 standard
deviations on either side of the mean This will mean that 99.7% of the bulbs will be
between 120 – 3×10 months and 120 + 3×10 So 99.7% of the bulbs will last 90-150 months This assumes that all the bulbs are produced
to the same standard
Example continued…
34% 34%
13.5% 13.5%
2.35% 2.35%
95%
99.7%
120 140 15010090months monthsmonthsmonths months
Exercises
try page 176 #1, 3b, 6, 8, 9, 10 http://onlinestatbook.com/
References
Lane, D. (2003). What's so important about the normal distribution? Retrieved October 5, 2004 from http://davidmlane.com/hyperstat/normal_distribution.html
Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page