Taming Statistics Move Cursor to tabs to Learn and Do Statistics.

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Taming Statistics: Learn & Do with Examples: by Stephen M Mansour & Jerry M Brennan.
Download any PDF listed below & then try examples below.
(A split screen with 2 windows might be helpful. Put PDF window on 1 side & this window on other side.)
(Then you can drag examples from PDF into Input: box below, and click Calc button.)
  • APL Lessons and Examples: PDF by Jerry M Brennan 67 pages (revised 4/19/2017)
  • Statistical Calculations by Stephen M. Mansour(pdf) (revised 1/1/2015)
  • Reference Card by Stephen M. Mansour(pdf) (revised 9/13/2016)
  • Inferential Statistics by Stephen M. Mansour(pdf) (revised Jan 20, 2015)
  • Taming Statistics Users Guide by Stephen M. Mansour(pdf) (revised 7/19/2016)
  • Click Here for: Users Guide SampleData.csv & choose where to save it on YOUR computer).
  • Now Import SampleData.csv variables into APL. Press Choose File below & then Press Import to Namespace D
    Now type/drag D.Student or D.Sex or.. to Input: box & Press Calc to see data.
    You can input data in 3 different ways.
    1. Data dragged or copied & pasted(i.e. from PDF examples) into the Input: box below.
    1b. Example: Drag or copy & paste +/1 3 4 to Input: box below & Press Calc
    1c. Example: Click any purple text +/2 4 6 7 to copy it to Input: box below & Press Calc
    2. Data from file: use Choose File button below. File can use either commas or spaces between values.
    2a. First row must contain variable names and following rows each contain 1 value for each of the named variables.
    2b. Variable names must begin with letter & contain only upper & lower case letters and/or digits 0-9. Spaces in names not allowed.
    2c. Data can be numeric or any characters(except commas). Character Spaces ok if comma separators.
    2d. Example: row 1 names: Name,Age rows 2&3 data:Jim Ito,12 and Jo Til,22
    2e. Data is stored in namespace: D so use full name to access it: D.Name or D.Age
    3. Data entered by you into Input: box below.
    3a. Begin with valid variable name(letter first & contain only upper & lower case and/or digits 0-9).
    3b. Variable name then assignment then data with spaces between #'s or chars with single quotes around each char data.
    3c. To get the assignment arrow to Input: box drag it or click Primer button then the arrow symbol .
    3d. Examples: Numeric: AGE←15 13 11 Character: SexOfChild←'Male' 'Unknown' 'Female'
    3e. Click on each of above in purple to copy to Input: box & press Calc on each.
    3f. Now Click each of these: AGE and SexOfChild to send to Input: box & press Calc on each.
    3g. Now Click each of these: ↑AGE SexOfChild and ⍉↑AGE SexOfChild to send to Input: box & press Calc on each.
    Central Tendency:(for ordered numeric data only)
    Four common measures of central tendency are: mean median mode proportion

    Let's create some random data and evaluate it.
    Click DATA←?25⍴9 & press Calc button below Input:
    Now click DATA & press Calc to see 25 random #'s from 1-9.
    Now click each then press Calc for each of the following: 1) mean DATA 2) median DATA 3)mode DATA
    Now create proportion data click: PRO←¯1+?35⍴2 & view it PRO & find it's central tendency proportion PRO
    Common measures of Dispersion:
    range min max variance standardDeviation quartile skewness kurtosis frequency
    All work on ordered numeric data. frequency also works on character & categorical numeric data.
    Try each on your data or from SampleData.csv(if variables listed below or Choose File to reimport them).
    For imported data from SampleData.csv remember to include Namespace D. Click: frequency D.Eyes
    For a labeled barchart click follwing to Input window and click Calc: FreqBar ⍉frequency D.Eyes
    Confidence Interval:
    If you took a # of samples from data pool & calculated means for each they would all be somewhat different. Each sample mean would be an estimate of the population mean. Most would be close to the population mean but some might differ quite a bit. If you took larger samples the sample means would tend to be closer to the population mean with fewer outlier means. Most would be either a little less or a little more with fewer extremely more or extremely less. Statistically this would tend towards a bell shaped curve. Statistics can thus be applied to compute a confidence interval. We can't be sure our sample is a perfect estimate, but we can determine a confidence interval around our sample mean using sample size & variability. A 95% confidence interval is often used(i.e. we're 95% certain the population mean is in the confidence interval around our sample mean.
    Example 1: You sample 100 students & 55 say they will vote for Mary. Can you be 95% certain she'll win(get >50% of votes)? Click purple code below to send code to Input: box & click Calc to see. Code below first creates the data & then determines confidence interval:
    .95 proportion confidenceInterval YES←55 45/1 0
    Now click it again but modify YES to 60 & no to 40 before pressing the Calc to see that the confidence interval.
    The confidence interval for 55/45 encompassed values below .50 so we could not be 95% sure, but 60/40 did not. Now play with bigger samples(try 200 ie 120 80 for the 60 40 %) or different confidence intervals such as .99 by making slight modifications of your Input: box below.
    Determine Sample Size Needed To Demonstrate Significant Results:
    If your samples are too small you will not demonstrate significant differences. If samples are too big you will be wasting time/money/your energy. Picking the right sample size if a very very important first step in doing research.
    To determine the sample size you must know the size of the difference you wish to detect and have an estimate of the variability you will encounter in your data. If you wanted to detect a 2 point difference of the mean in your sample data 95% of the time and the standard deviation in your data was 12.
    Click purple code to send the following to Input line below: .95 mean sampleSize 2 12 and press the Calc button to find the sample size needed.
    To be 99% sure that Mary will get >50% of the vote if in your sample of students 55% or more said they would vote for Mary, how big of sample do you need?
    Click and press calc on this to find the sample size needed: .99 proportion sampleSize .05 .55
    Hypothesis Testing: Confidence Intervals & t-test
    If average height of people is 68", is that significantly different than SampleData.csv data?
    Significantly commonly means we want to be 95% certain difference we got is not due to chance.
    Since data is a sample of people other samples could be taken & they would have different means.
    Let's calculate sample mean & using sample variability the 95% confidence interval of the mean.
    Click following purple code to Input: box and press Calc
    mean D.Height and .95 mean confidenceInterval D.Height
    Since the 95% confidence interval does not include 68 we are 95% sure that sample is different.
    We can also do .05 significance test: .05 report D.Height mean hypothesis = 68
    Now to see if male heights differ significantly from females: Click and Calc each of next 3 lines
    mean MaleHeight←(D.Sex eq'M')/D.Height
    mean FemaleHeight←(D.Sex eq 'F')/D.Height
    .95 mean confidenceInterval¨ FemaleHeight MaleHeight
    No overlap in confidence intervals so 95% sure male & female means are not equal(ne).
    Now we can verify with this: .05 report MaleHeight mean hypothesis ne FemaleHeight
    Hypot Test:paired t-test
    On previous tab we compared 2 different groups: males and females. In this example we compare same people before and after a treatment. In this case imagine the same students being tested before and after watching a calculus video. Click on each purple text below to copy to Input: then press Calc to see each result.
    mean PRE←18 21 16 22 19 24 17 21 23 18 14 16 16 19 18 20 12 22 15 17
    mean POS←22 25 17 24 16 29 20 23 19 20 15 15 18 26 18 24 18 25 19 16
    DIFS,mean DIFS←POS-PRE ⍝ show diffs & mean dif between pre & post scores for each student.
    .05 report DIFS mean hypothesis>0 ⍝ Average difference is significantly > 0 (1tailed t-test ⍺=.05)
    .95 mean confidenceInterval DIFS ⍝ 95% confidence interval of average diff(2.05) is above 0.
    click: report regress D.ShoeSize D.Height D.Student
    Results:B0 intercept significant p=.00002 (but not really important avg ShoeSize is < Height)
    Results:B1 significant p=.00000 (Height is useful in predicting ShoeSize)
    Results:B2 not significant p=.71269 (Student # is not useful in predicting ShoeSize)
    click: D.ShoeSize RegLinPlot D.Height
    Plot:There are 100's of plotting routines. Here are a few you may find useful.
    plotxy for example plots first variable on x and all rest(1 or more) on y axis:
    click: X←1 2 3 4 ⋄ Y1←4 6 7 8 ⋄ Y2←3 5 9 11 ⋄ plotxy X Y1 Y2
    click: ch.Set('Head' 'My Plot')('Ycap' 'Y label')('Xcap' 'Xlabel')('Key' 'Y1,Y2') ⋄ plotxy X Y1 Y2

    click: stemleaf 500+?50⍴120
    click: boxandwisker 5+?50⍴12
    click: FreqBar Freq ?500⍴12 or just click: freq ?500⍴12 to see just frequency table.
    click: QuadPlot 2 ¯1 ¯7 ⍝ for Equation: (2x*2)+(¯1x)+(¯7) and thus:a=2 b=¯1 c=¯7
    click: 5 25 .000001 PlotAreaUnderCurve '(X*2)+5' ⍝ for Y=5+X*2 from X=5 to 25 with 6 decimal accuracy

    Click on Primer button to show/hide the APL Primer a floating keypad of APL symbols.

    Drag the floating keypad to a convenient place on the screen.

    Hover mouse over any APL character to see a pop-up explaining more about the symbol with examples and shortcut to enter it directly from your keyboard.

    If you "click" on any APL symbol on Keypad that character will be added to the INPUT field below.

    When your INPUT field is completed Click CALC button or press your "ENTER KEY" to process your calcuation and see the results.

    To input datafile (row1=var names, rows2-n=data) Choose File then Import
    Input :
    Result: Set Rows Visible for Scrollable Results Window=