(a.k.a. Mean average)
Chapter 5
Describing Distributions Numerically
1
Mean is the sum of the measurements divided by the number of observations (n) in the data set n Formula:
y=
i =1
yi
n
What does the mean really measure?
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Chapter Objectives
Discuss various numerical measures of center and spread Identify the basic properties of these measures Construct box plots to summarize distributions Compare distributions using box plots Identify procedures for describing distributions numerically in SPSS
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Calculating the Mean
During the first ten basketball games of the season, ISUs points leader had the following points: 29,28,29,31,30,31,27,29,30,32
y= 29 + 28 + 29 + 31 + 30 + 31 + 27 + 29 + 30 + 32 = 29.6 10
U of Is points leader put up the following numbers: 40,28,29,31,30,31,27,29,39,36
y= 40 + 28 + 29 + 31 + 30 + 31 + 27 + 29 + 39 + 36 = 32.0 10
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Measures of Center
Mean Median Mode Midrange
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Mean as the Balance Point
ISU Player
. .
27.5
. :
30.0
: : . .32.5
Points scored 37.5 40.0
Mean=29.6 Points
U of I Player
. . :
27.5
. :
30.0 32.5
.
37.5 Mean = 32.0 Points
. :
40.0
Points scored
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Median
The median is the value of the middle observation in an ordered set of data.
Example: The following data represent the ages of motorcyclists when they were fatally injured in traffic crashes : 17,38,27,14,18,34,16,42,28,20,23,21,24
Why Use Two Different Measures?
!"
Do the results support the common belief that such fatalities are incurred by younger drives. Calculate the median of the data to answer this question.
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# %
# $ " $
$
10
Calculating the Median
Step 1: Order the data from lowest to highest
Location Age 1 14 2 16 3 17 4 18 5 20 6 21 7 23 8 24 9 27 10 28 11 34 12 38 13 42
Comparing the Mean and the Median
If Mean = Median Mean < Median Distribution is Symmetric Skewed left Skewed right
Step 2: Locate the middle observation. Since n=13 is odd, the middle position is (13+1)/2 = 7th observation. Median is the value of the 7th observation = 23.
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Mean > Median
Rule of Thumb - if the mean and median differ by more than 10% the data are skewed. Warning - does not hold in all cases Draw a picture!
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Calculating the Median (cont)
What if we have an even number of obs.?
Location Value 1 10 2 11 3 14 4 15 5 18 6 19 7 22 8 23 9 25 10 85
Other Measures of Center
Mode - the value that occurs most often in the data set.
Might not exist or there may be more than one mode More appropriate for categorical data
When n is even the median is the average of the observations in the middle positions, n/2 and n/2+1. So we find the median is (18 + 19)/2 = 18.5
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Midrange = highest value + lowest value 2
Very sensitive to extreme values
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2
Measures of Spread
Range Interquartile Range Standard Deviation Variance
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Steps for Finding Quartiles
Put the data values in order and find the median Take the lower half of the data (values less than the median) and find the median of the lower half of the data. This values is the first quartile. Q1 Take the upper half of the data (all of the values greater than the median) and find the median of the upper half of the data. This value is the third quartile. Q3 Note: When n is odd, include the median in 16 both halves of the data
Range
The range is the difference between the maximum and minimum values in the data. Range = Max - Min
Location Value 1 10 2 11 3 14 4 15 5 18 6 19 7 22 8 23 9 25 10 85
Quartile Examples
n=8 45 Lower Half 48 52 54 55 Upper Half
Upper Half n=9 67 Lower Half 70 71 74 75 76 Median Q1
Q1=(48+52)/2=50
58 62 64 Q3(58+62)/2=60
77 77 79
Q3
Range = 85 - 10 = 75 units
Advantage - easy to calculate and understand Disadvantage - sensitive to outliers
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Interquartile Range (IQR)
Q1 (first quartile) separates - the bottom 25% of the sorted data values from the top 75% Q3 (third quartile) - separates the bottom 75% of the sorted values from the top 25%. IQR = Q3 - Q1
Identifies the range of the middle half of the data Is not sensitive to outliers Should be used as the measure of spread 15 when data is skewed
Standard Deviation
The standard deviation of a set of sample values is a measure of spread of values about the mean. It is a type of average deviation of values from the mean that is calculated by using the following formula:
s=
( y y )^ 2 n 1
Is the most important and useful measure of spread.
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3
Standard Deviation Example
The Normal town council is studying a traffic flow problem. The following data represent the number of cars that drove through a particular intersection during a 5-minute period of the course of 6 days.
Day 1 2 3 4 5 6 Cars 20 27 29 28 37 23
Which of These Distributions Has the Larger Standard Dev.?
See Distributions Presented In Class
Calculate the standard deviation of the number of cars that drove through the intersection.
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Calculating the Standard deviation
1. 2. 3. 4. 5. 6.
Displaying Distributions Using Boxplots
120000
156
Calculate the mean of the data Find the distance of each data value from the mean Square each of the differences found in step 2 Add up the squared deviations Divide value found in step 4 by n - 1 (This is the variance of the data) Take the square root of the variance
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Solid black line represents the median
2 6
100000
80000
Height of the box is equal to the IQR Outliers are displayed individual Placement of median in the box suggests something about the shape of the distribution
60000
40000
20000
N= 207
Faculty salary ($ pe
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Properties of the Standard Deviation
Measures the variation of all values from the mean The larger the s.d. the larger the spread Sensitive to extreme values Use for symmetric distributions Usually >0. Only = 0 when all data values are the same Reported in the same units as the original data
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Procedure for Constructing a Boxplot
1.
2. 3. 4.
Find the 5-number summary consisting of the minimum value, Q1, median, Q3 and the maximum value. Construct a scale with the values that include the minimum and maximu...