Measures of Spread

Chapter 3.3 – Tools for Analyzing Data

I can: calculate and interpret measures of spread

MSIP/Home Learning: p. 168 #2b, 3b, 4, 6, 7, 10

What is spread?

Measures of central tendency do not always tell you everything!

The histograms have identical mean and median, but the spread is different

Spread tells you how widely the data are dispersed

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Why worry about spread? Spread is how closely the values cluster

around the middle value Less spread means you have greater

confidence that values will fall within a particular range

Important for making predictions

Measures of Spread There are 3 Measures of Central Tendency:

Mean, Median, Mode All measure the centre of a set of data We will also study 3 Measures of Spread:

Range, Interquartile Range, Standard Deviation

All measure how spread out data is Smaller value = less spread, more consistent

Measures of Spread 1) Range = (max) – (min)

Indicates the size of the interval that contains 100% of the data

2) Interquartile Range IQR = Q3 – Q1 where Q1 is the lower half median and Q3 is the

upper half median Indicates the size of the interval that contains the

middle 50% of the data

Quartiles Example 26 28 34 36 38 38 40 41 41 44 45 46 51 54 55

Q2 = 41 Median Q1 = 36 Lower half median Q3 = 46 Upper half median IQR = Q3 – Q1 = 46 – 36 = 10 (50% of data is 10 units apart) If a quartile occurs between 2 values, it is

calculated as the average of the two values

A More Useful Measure of Spread Range is very basic

Does not take clusters or outliers into account Interquartile range is somewhat useful

Takes clusters and outliers into account Visual in Box-and-Whisker Plot

Standard deviation is very useful The average distance from the mean for all data

points

Standard Deviation

1. Find the mean (average)

2. Find the deviation for each data point

(data) – (mean)

3. Square the deviations

4. Average the squares of the deviations (this is called the variance)

5. Take the square root of the variance

Deviation The mean of these numbers is 48 Deviation = (data) – (mean) 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

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 = 25 + 9 + 9 + 25

4 σ² = 17 σ = √17 = 4.1

Measure of Spread - Recap Measures of Spread are numbers indicating how spread

out data is Smaller value for any measure of spread means data is

more consistent 1) Range = Max – Min 2) Interquartile Range: IQR = Q3 – Q1

Q1 = first half median Q3 = second half median

3) Standard Deviation Find mean (average) Find all deviations (data) – (mean) Square all and average them - this is variance or σ2

Take the square root to get std. dev. σ

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

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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 = 0.9

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Frequency 2 6 6 2

MSIP / Home Learning 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 (~6 pieces of data) We will use technology (Fathom/Excel) to

calculate larger examples Have a look at your calculator and see if you

have this feature (Σσn and Σσn-1)

Normal Distribution

3.4 – Tools for Analyzing Data

Learning goal: Determine the % of data within intervals of a Normal Distribution

Due now: p. 168 #2b, 3b, 4, 6, 7, 10

MSIP / Home Learning: p. 176 #1, 3b, 6, 8-10

Histograms

Histograms can be skewed...

Right-skewed Left-skewed

Histograms

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Normal? A normal distribution is a histogram that is

symmetrical and has a bell shape Used quite a bit in statistical analysis Also called a Gaussian Distribution 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

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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

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 with:

a) 68% confidence? b) 95% confidence? So X~N(120,102)

An example cont’d

68% of the data will be within 1 standard deviation of the mean

This will mean that 68% of the bulbs will be between

120–10 = 110 months and 120+10 = 130 months So 68% of the bulbs will last 110 - 130 months

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 = 100 months and 120 + 2×10 = 140 months 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

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: height, reading ability, memory, IQ, etc.

Normal distributions are statistically easy to work with All kinds of statistical tests are based on it

Lane (2003)

Exercises

Complete p. 176 #1, 3b, 6, 8-10 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