Download Intro to Statistics

LSP 121
Week 2
Intro to Statistics and SPSS/PASW
Descriptive Statistics:
Mean, Median, Percentile, Range
• Mean
• Median – the middle score
• The score with an equal number of data points above and below
• If there are an even number of datapoints, take the average of the
middle two
• Percent Rank – calculates the position of a datapoint in a data
set. More precisely, tells you approximately how many
percent of the data is less than the datapoint.
• e.g. 86th percentile means that 86 percent of data-points /people / etc
were below that number
• Range – difference between the maximum and minimum
values in the data set
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Median
• Median for bank 1 = the middle value of 11
data points
• Median for bank 2: even number of data
points – there is no middle.
– Take the average of the two middle values
Bank 1:
4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0
Bank 2:
6.6 6.7 6.7 6.9 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8
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Descriptive Statistics: Quartiles
• Lower quartile: aka first quartile - the median
of the data values in the lower half of a data
set (do not include the median)
• Middle quartile: aka second quartile - this is
the overall median
• Upper quartile: aka third quartile - the median
of the data values in the upper half of a data
set (do not include the median)
– Note: Some statistical software packages use the 25th, 50th,
and 75th percentiles as their quartiles (instead of median
values). SPSS determines quartiles in this way. On an
exam, you would use the medians.
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Quartiles
• For example (bank waiting times):
lower quartile
median
upper quartile
Bank 1:
4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0
Bank 2:
6.6 6.7 6.7 6.9 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8
Bank 2
median = (7.1 + 7.2)/2 = 7.15
lower quartile = 6.7
upper quartile = 7.7
range: 7.8 – 6.6 = 1.2
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Descriptive Statistics:
The Five-Number Summary
• The five number summary consists of:
– The minimum value
– The lower quartile (first quartile)
– The median (second quartile)
– The upper quartile (third quartile)
– The maximum value
• As mentioned earlier, SPSS determines quartiles using the percentiles:
First quartile is 25th percentile, second quartile is 50th percentile, and third
quartile is 75th percentile
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Standard Deviation
• Quartiles are OK for characterizing data, but
standard deviation is preferred by statisticians
• It is a measure of how far data values are
spread around the mean of a data set
• Formula:
– Std dev = sqrt(sum of (deviations from the mean)2
/ total number of data values – 1)
– You don’t need to know this formula!
– Don’t calculate by hand, use statistical software
such as SPSS (which we’ll do in a few minutes)
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Standard Deviation - Guesstimate
• A simple way to estimate standard deviation is the range
estimate
• Don’t rely on estimation – use only to get a very quick and
general idea of the value of sd.
• Divide range by 4
• Watch for outliers. They can ruin your range estimate
• What is an outlier?
• Two or more standard deviations from the mean (above OR
below)
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Standard Deviation
• Go back to Big Bank / Best Bank example
• Big Bank: range = 6.9
• 6.9 / 4 = 1.7
• Actual standard deviation is 1.96
• Best Bank: range = 1.2
• 1.2 / 4 = 0.3
• Actual standard deviation is 0.44
• Any outliers? Means are 7.2 and 6.7
Big Bank:
4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0
Best Bank:
6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8
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* Histograms
• Nice way to view a data set
• A histogram is a chart created by defining a
set of bins and counting how many data points
lie in each bin. Bars are drawn with height
proportional to the number of data points in
each bin.
– * Note: The histogram does not keep track of the
value of each data point – it only keeps track of
which bin a data point is contained in.
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Example Histogram
Salaries of 26 Men’s Basketball Coaches
What is the most common salary according
to this graph? How many coaches make this
amount?
Between $50,000 and $100,000
Most of the coaches (15).
How many coaches make less than
$50,000?
Only 1.
How many make more than than $100,000?
About 10.
These would make for good exam questions…
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Statistics and SPSS/PASW
• While Excel can do some basic statistics, it is
not considered a serious statistics tool
• You really should use something like
SPSS/PASW or SAS
• We’ll use SPSS/PASW since DePaul has a site
license
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Let’s Try An Example
• Copy the dataset grades.xls (from the QRC web page  Excel Files  Older Data) to My
Documents and start SPSS
• or try the file IncomeGaps.xls
• Open the Grades.xls spreadsheet
• Note: SPSS looks for files with an extention of .sav However, Excel files
have an .xls extension. You must select the ‘Files of Type’ dropdown to tell
SPSS to search for XLS (i.e. Excel) files.
• Change the variable names and make sure the data is numeric, not text
• Click on the ‘Variable View’ tab at the bottom
• For each of the two rows, click the cell under ‘Type’ and choose Numeric.
• Then click back to ‘Data View’
• Click on Analyze -> Descriptive Statistics -> Frequencies
• Copy any variables that you want to analyze (i.e. exam 1 and exam 2) into
the box on the right
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Let’s Try An Example
• Be careful! If the numeric fields in the dataset
have any $, % or #, SPSS will have difficulty
converting these to numeric
• In particular, if the data has dollar signs, have
SPSS first convert the field to Dollar, then
convert it to Numeric (IncomeGaps.xls)
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Let’s Try An Example
• Using the grades for Exam 2, find the
– 5 number summary (minimum, 1st quartile,
median, 3rd quartile, maximum)
• See this link for instructions
– Mean
– Range
– What is the standard deviation?
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Listing Z-Values
• A good stats package will make
it easy to determine z-values
• Click on Analyze  Descriptive
Statistics  Descriptives
• Choose the variable, let’s use
Exam2
• Be sure the check ‘Save
standardized values as
variables’ at the bottom
• When you return to the ‘Data
View’ you will see that a new
column has appeared giving
you the z-score for every value
in the Exam2 data set
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Pivot Tables
• Let’s say you have just performed a survey.
• One of the questions you ask is: “What type of
home computer Internet connection do you
have?”
– Answers can be: None, Dial-up, DSL, Cable, Other,
Not Sure.
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Pivot Tables
• Here are some of your results
Respondent ID
11111
11112
11113
11114
11115
11116
Cable Type
no
ds
cm
dk
du
du
Where no = none; ds = dsl; cm = cable modem;
du = dial up; dk = don’t know; ot = other
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Pivot Tables
• You can use SPSS to count the occurrences of
data items, just like a pivot table
• Open a new file: File  New
• Enter your data into SPSS (you can leave out the IDs
for now)
• Click on Analyze / Descriptive Statistics / Frequencies
• Move the variable that you want to count from the
left box to the right box
• Make sure Display Frequencies Table is checked
• Run it (Click ‘OK’)
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Crosstabulations
(Crosstabs)
• Crosstabs are an extension of pivot tables
• Let’s say you have asked a number of
students: How many schools did you apply to?
• You get results something like the following (in
a spreadsheet):
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Crosstabs
Respondent ID
Sex
# of schools
1
F
6
2
M
2
3
F
7
4
M
4
5
F
9
6
F
10
7
M
3
8
M
2
9
F
7
10
F
5
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Crosstabs
• Now open the data in SPSS
• Then pull down the menu Analyze and click on
Descriptive Statistics, then Crosstabs
• What variable do you want in the row? The
column?
– We are probably interested in determining
examining how many schools females apply to
relative to males
• When ready, click OK to perform the crosstab.
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