1. Frequencies and the normal distribution
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2. Make a histogram with a bin width of 0
Make a histogram with a bin width of 0.1 using the House Fly data provided.
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3. Is that normal?
We want to compare the distribution to a normal distribution.
We will make a histogram is a more tedious procedure that will allow us to also plot the normal distribution function for a nice visual comparison.
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4. Calculate the mean (average) and standard deviation of the House –fly wing data
The normal distribution uses the average and standard deviation as parameters controlling the center of the distribution and how spread out it is.
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5. Calculate the sample size (count) of the data
The normal distribution is about probabilities. We will use the count to turn a frequency (the number of times something occurred) into a percentage/probability that something occurred.
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6. Calculate the minimum and maximum of the data
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7. Make a range of data starting at the Min plus 0
Make a range of data starting at the Min plus and going to the Max by 0.1’s
We are making the bins that Excel generated so nicely for us in the fast histogram.
There is no exact rule for what the bin width should be – that’s why Excel lets you adjust that parameter.
Except in the tail of the distribution, you probably don’t want frequencies of 1 or 0.
We are using a bin width of 0.1
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8. Warning: Array formula
Array formulas have results that span several cells instead of just one.
They require:
Highlighting enough cells for the complete answer
Clicking Ctrl-Shift-Enter instead of just Enter.
If you attempt to edit part of an array formula result, problems occur. Use the Esc key to get out of it.
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9. Highlight cells D2:D21 (for the answer) and insert the formula =FREQUENCY(A2:A101,C2:C21) and then hit Ctrl-Shift-Enter
The range A2:A101 is the data and the range C2:C21 are the bins.
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10. A frequency is the number of times something occurs – in this case the values up to and including 3.6, then the values between 3.6 (exclusive) and 3.7 (inclusive), etc.
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11. Make a quick (unformatted, un-designed) XY Scatter graph
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12. Normal distribution
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13. Use Excel’s formula for the normal distribution
Notice that we have used absolute addressing for the mean B$2 and standard deviation B$4. That way when the formula is copied elsewhere, the 2 and 4 are held fixed.
The 4th argument is whether or not we want to sum the distribution everything from negative infinity up to and including a value – we said no.
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14. Before comparing we need two more steps
Before comparing we need two more steps. The first is to divide the frequencies by the sample size turning them effectively into probabilities
Instead of saying the value 4.2 occurred 7 times in a sample size of 100, we say 4.2 occurred 0.07 or 7% of the time.
Note the count requires absolute addressing.
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15. The normal distribution has to be multiplied by the bin width – the separation between our values
We had gone up by 0.2’s instead of by 0.1’s there would be roughly half as many Frequencies with roughly twice the value they have now.
In math class we’d probably call the bin width Δx. This way we can get a single point to “stand in for” all the values from x to x+ Δx
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16. Highlight columns C, F & G (only where there’s data) and Insert an XY-Scatter chart.
To highlight non-consecutive columns: highlight the first column then hold down the Ctrl key while highlighting the second (and third) column. Then let go of Ctrl.
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17. Apply a Chart Layout (e.g. 1) under Design.
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18. Change the title and axis labels
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19. Format the axis to have a Minimum of 3
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20. Right click on a data point and choose Select Data
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21. Select a data set, click Edit and give the series a name.
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22. Result – see legend
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23. Right click on the second (normal dist
Right click on the second (normal dist.) series and choose Change Format Data Series. Choose the Paint bucket (Fill & Line). Choose Line, Solid Line and pick a color
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24. And on Marker choose None
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25. Result
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26. Skew and Kurt
Recall that the skew (skewness) and kurt (kurtosis) calculations also provide measures of how close a distribution is to normal or in what way and by how much it deviates from a normal distribution.
Calculate skew and kurt for the fly data
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27. Skewness rule of thumb (https://brownmath.com/stat/shape.htm)
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29. Histogram alternative
The rest is how to make a histogram using the frequency data we just generated.
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30. Histogram: highlight columns C & D and choose Insert Column Chart
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31. Right click and choose Select Data
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32. Highlight Series 1 and click Remove
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33. Highlight Series 2 and click Edit under Horizontal Axis Labels
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34. Then highlight the C column and click OK
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35. Right click on the x axis, Choose Format, choose Number, change the category to Number and set the Decimal place to 1
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36. Result so far
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37. Choose a Layout (e.g. 7) and label axes
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38. Add a title (if you haven’t got one already)
Add a title (if you haven’t got one already). Design tab, Add Chart Element, …
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39. Right click on the columns and choose Format Data Series
Right click on the columns and choose Format Data Series. Choose the gap width to be 0%.
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40. Choose a Solid Line and a Border Color.
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41. Result
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