Embed Size (px)
Citation preview
1
Describing distributions with numbers
William P. Wattles
Psychology 302
2
Measuring the Center of a distribution
Mean– The arithmetic average– Requires measurement data
Median– The middle value
Mode– The most common value
3
Measuring the center with the Mean
xX
n
4
Our first formula
X the mean
X the individual score
n the number of individuals
sum of
5
The Mean
One number that tells us about the middle using all the data.
The group not the individual has a mean.
Population
Sample
6
Sample mean
X
7
Mu, the population mean
Population
Sample X
8
Calculate the mean with Excel
Save the file psy302 to your hard drive– right click on the file – save to desktop or temp
Open file psy302 Move flower trivia score to new sheet
9
Calculate the mean with Excel
Rename Sheet– double click sheet tab, type flower
Calculate the sum– type label: total
Calculate the mean– type label: mean
Check with average function
10
Measuring the center with the Median
Rank order the values If the number of observations is odd the
median is the center observation If the number of observations is even
the median is the mean of the middle two observations. (half way between them)
11
Measuring the center with the Median
Mediann
1
2
12
The mean versus the median
The Mean– uses all the data– has arithmetic properties
The Median – less influenced by Outliers and extreme
values
Mean vs. MedianBetty 28,514.00$ Mike 22,316.00$ Tom 30,112.00$ Miriam 29,521.00$ Stacy 21,555.00$ David 125,366.00$ Mary Lou 22,132.00$ John 27,561.00$ Gail 24,635.00$ Arthur 30,125.00$
mean 36,183.70$ median 28,037.50$
5
The Mean
The mean uses all the data. The group not the individual has a
mean. We calculate the mean on
Quantitative Data
Three things to remember
The mean tells us where the middle of the data lies.
We also need to know how spread out the data are.
Mean = 68 inches Mean = 68 inchesStd Dev= .57 inchesStd Dev= 8.54 inches
Measuring Spread
Knowing about the middle only tells us part of the story of the data.
We need to know how spread out the data are.
Variability
Variety is the spice of life
Without variability things are just boring
exam3 Psy314 Health Psychology69% 61% 79% 100%54% 60% 85% 83%58% 75% 85% 73%87% 57% 80% 83%65% 68% 58% 50%83% 55% 59% 79%89% 74% 85% 63%
Why is the mean alone not enough to describe a
distribution?Outliers is NOT the answer!!!!
The mean tells us the middle but not how spread out the
scores are.
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
14
Example of Spread
New York mean annual high
temperature 62
14
Example of Spread
San Francisco mean annual high
temperature 65
16
Example of Spread
New York mean max min range sd 62 84 39 45 17.1
San Francisco 65 73 55 18 6.4
Example of Variability
Psy 302 Spring 2003
0%
20%
40%
60%
80%
100%
120%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Student
Gra
de Final
Quiz 7
17
Measuring Spread Range Quartiles Five-number summary
– Minimum– first quartile– median– third quartile– Maximum
Standard Deviation
Mean 50.63%
Mean 33.19%
Std Dev 21.4%
Std Dev 13.2%
19
Deviation score
Each individual has a deviation score. It measures how far that individual deviates from the mean.
Deviation scores always sum to zero. Deviation scores contain information.
– How far and in which direction the individual lies from the mean
18
Measuring spread with the standard deviation
Measures spread by looking at how far the observations are from their mean.
The standard deviation is the square root of the variance.
The variance is also a measure of spread
Mean Xdeviation score
$28,756 32,092$ The average teacher $28,756, John $32.092 64.5 68 The average woman is 5 4 1/2, mary is 5'8"110 90 The average IQ is 110 and Bubba has a 90
Individual deviation scores
deviation score
The average teacher $28,756, John $32.092 $3,336 dollarsThe average woman is 5 4 1/2, mary is 5'8" 3.5 inchesThe average IQ is 110 and Bubba has a 90 -20 points
Standard deviation
One number that tells us about the spread using all the data.
The group not the individual has a standard deviation.
Note!!
23
Standard Deviation
sx x
n
( ) 2
1
22
Variance
sx x
n2
2
1
( )
24
Properties of the standard deviation
s measures the spread about the mean s=0 only when there is no spread. This
happens when all the observations have the same value.
s is strongly influenced by extreme values
New Column headed deviation Deviation score = X – the mean
25
Calculate Standard Deviation with Excel
In new column type heading: dev2 Enter formula to square deviation Total squared deviations
– type label: sum of squares Divide sum of squares by n-1
– type label: variance
Moore page 50Example 2.7page 50
subject MetabolicRatesubject1 1792subject2 1666subject3 1362subject4 1614subject5 1460subject6 1867subject7 1439
Example 2.6page 42
subject MetabolicRatedev dev2subject1 1792 192 36864subject2 1666 66 4356subject3 1362 -238 56644subject4 1614 14 196subject5 1460 -140 19600subject6 1867 267 71289subject7 1439 -161 25921
total 11200 0 214870 ssmean 1600 35811.6667 var
189.239707 stdev189.239707 stdev check
To Calculate Standard Deviation: Total raw scores divide by n to get mean calculate deviation score for each subject (X
minus the mean) Square each deviation score Sum the deviation scores to obtain sum of
squares Divide by n-1 to obtain variance Take square root of variance to get standard
deviation.
Population
Sample
26
Sample variance
s2
27
Population variance
2
Population Variance
Sample Variance
2
s2
28
Little sigma, thePopulation standard deviation
29
Sample standard deviation
s
Population Standard Deviation
Sample StandardDeviation
s
To analyze data
1. Make a frequency distribution and plot the data
Look for overall pattern and outliers or skewness
Create a numerical summary: mean and standard deviation.
41
Start with a list of scoresCathy 400 Alice 300Paula 300 Mitzi 200Sandy 500 Jack 700Lois 400 Mike 500Anne 500 Dawn 600Miriam 600 Vicki 400June 400 George 500David 500 Ashley 800
42
Make a frequency distribution
200 xxxxxxxxxxx300 xxxxxxxxxxxxxxxxxxxxxxxx400 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx500 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx600 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx700 xxxxxxxxxxxxxxxxxxxxxxxxx800 xxxxxxxxxxxxx
43
Frequency distributionscore frequency
200 50300 100400 150500 250600 150700 100800 50
44
Represent with a chart (histogram)
SAT Scores
0
50
100
150
200
2502
00
30
0
40
0
50
0
60
0
70
0
80
0
Score
Fre
qu
en
cy
45
Represent with line chartSAT Scores
0
50
100
150
200
250
20
0
30
0
40
0
50
0
60
0
70
0
80
0
Score
Fre
qu
en
cy
Density Curve
Replaces the histogram when we have many observations.
Transform a score
Hotel Atlantico 200 pesos Peso a unit of
measure
Transform a score
1 dollar = 28.38 pesos
200/28.38=$7.05 Dollar a unit of
measure
31
standardized observations or values. To standardize is to transform a score
into standard deviation units. Frequently referred to as z-scores A z-score tells how many standard
deviations the score or observation falls from the mean and in which direction
32
Standard Scores (Z-scores)
individual scores expressed in terms of the mean and standard deviation of the sample or population.
Z = X minus the mean/standard deviation
33
Z-score
zx
34
new symbols
the population mean
the population s dard deviation
z the s dardized value
tan
tan
35
Calculate Z-scores for trivia data
Label column E as Z-score Type formula deviation score/std dev Make std dev reference absolute (use
F4 to insert dollar signs) Copy formula down. Check: should sum to zero
File extensions
Word .doc Excel .xls Text files .txt
To view File extensions
Open Windows Explorer Choose Tools/Folder Options/View uncheck “hide extensions for known file
types.
37
Z Scores
Height of young women– Mean = 64– Standard deviation = 2.7
How tall in deviations is a woman 70 inches?
A woman 5 feet tall (60 inches) is how tall in standard deviations?
38
Z scores
Height of young women– Mean = 64– Standard deviation = 2.7
How tall in deviations is a woman 70 inches? z = 2.22
A woman 5 feet tall (60 inches) is how tall in standard deviations? z = -1.48
39
Calculating Z scores
48.17.2
6460
height
z
22.27.2
6470
height
z
Calculating X from Z scores
*zX
72
Types of data
Categorical or Qualitative data– Nominal: Assign individuals to mutually
exclusive categories. exhaustive: everyone is in one category
– Ordinal: Involves putting individuals in rank order. Categories are still mutually exclusive and exhaustive, but the order cannot be changed.
73
Types of data
Measurement or Quantitative Data– Interval data: There is a consistent interval
or difference between the numbers. Zero point is arbitrary
– Ratio data: Interval scale plus a meaningful zero. Zero means none. Weight, money and Celsius scales exemplify ratio data
– Measurement data allows for arithmetic operations.
60
The End
Mean vs. Median
