Download 2.5: Measures of Center

CHS Statistics
2.5: Measures of Spread
Objective: To calculate the standard deviation, IQR,
and range of data and determine the most appropriate
measure of spread for different sets of data
Warm-Up
 You are trying to decide which Biology class to
take. You have 2 instructors to choose from.
Professor Smith’s classes have a mean score of
82% with scores varying from 60% to 95%.
Professor Jones’ classes have a mean score of
85% with scores varying from 78% to 90%. Who
would you choose?
 You are trying to figure out which brand of light
bulbs to buy. Both have a mean lifetime of 1100
hours. Which would you buy? Given this scenario
you may purchase the cheaper of the two.
However, if you knew the GE bulb’s lifetime varies
from 500 to 1500 hours and the Walmart brand
bulb’s lifetime varies from 1000 to 1200 hours
would you still buy the cheaper one?
4 Measures of Variation (Spread)
1. Range
2. Interquartile Range
3. Variance
4. Standard Deviation
4 Measures of Variation (Spread)
1. Range – the distance between the maximum and
minimum values
Range = Max Value – Min Value
 Example: 1
2
 Example: 1
18 19 20 20 20 20
Affected by outliers?
5
8
12 16 20
4 Measures of Variation (Spread)
2. Interquartile Range – The distance between the
first and third quartiles
IQR = Q3 – Q1
 Contains the middle 50% of data
Q1 – One quarter (25%) of the data lie
below the first quartile
Q2 – the median
Q3 – one quarter (25%) of the data lie above
the third quartile
4 Measures of Variation (Spread)
2. Interquartile Range
 Example: Find the IQR of the set of data:
1
18 19 20 20 20 20
Affected by outliers?
4 Measures of Variation (Spread)
2. Interquartile Range
 You can find the quartiles in the graphing calculator
just like you would find the mean. Scroll down and
find Q1 and Q3.
 Enter your data into L1 (Stat  Edit)
 To clear data, be sure to highlight the title
(L1) and press CLEAR  Enter (DO NOT
PRESS DELETE)
 Once your data is entered, press STAT 
CALC  1-Variable Stat
4 Measures of Variation (Spread)
3. Variance
 The variance, notated by s2, is found by
summing the squared deviations and (almost)
averaging them:
Sample Variance:
Population Variance:
𝟐
(𝒙
−
𝝁)
𝝈𝟐 =
𝑵
 Why are we ALMOST averaging them?
 The variance will play a role later in our study,
but it is problematic as a measure of spread—it
is measured in squared units!
4 Measures of Variation (Spread)
4. Standard Deviation - the square root of the
variance and is measured in the same units
as the original data
 The standard deviation measures how far
each value is from the mean.
 Unless otherwise specified, assume the
data collected are a sample and find the
sample SD.
Sample Standard Deviation:
Population Standard Deviation:
𝝈=
(𝒙 − 𝝁)𝟐
𝑵
4 Measures of Variation (Spread)
4. Standard Deviation
Steps for Calculation:
1) Find the sample mean.
2) Subtract the sample mean from each value of
data.
3) Square the difference from #2.
4) Find the sum of the values from #2.
5) Plug into the formula.
Affected by outliers?
4 Measures of Variation (Spread)
4. Standard Deviation
Example: Points scored Week 1 NFL Winners.
Points
9
10
16
17
20
21
23
24
24
27
4 Measures of Variation (Spread)
4. Standard Deviation
 Steps on the calculator: Enter your data
into L1 (Stat  Edit)
 To clear data, be sure to highlight the
title (L1) and press CLEAR  Enter (DO
NOT PRESS DELETE)
 Once your data is entered, press STAT 
CALC  1-Variable Stat
 Use sx for the sample standard deviation
and 𝝈𝒙 for population standard deviation.
Rule of Thumb for Spread
Look at a histogram of your data to know
the best measure of spread to report!
 When the histogram of your data is fairly
symmetric, report the standard deviation,
because it is a more accurate measure of
spread.
 When the histogram of your data is skewed
in any direction, report the IQR as a more
appropriate measure of spread. Why?
Assignment
2.5 Worksheet