Module I3 Sessions 6&7
Practical 3: Processing multiple variables
This practical uses both CAST and Excel.
Overall Summary statistics in Excel

Go into Excel and open the workbook called summary statistics.xls. Go to
the sheet, called ANOVA. The data are the same as were used in practical 2, in
the sheet called Rice Yields.

The mean has been given, in cell B39. Use the Excel functions from the last
practicals to give the corrected sum of squares in B40, the d.f. (i.e. n-1) in B41,
the variance in B42, and the standard deviation in B43.

Use the 70-95-100 rule of thumb to state roughly what you know about the rice
yields of farmers:

Examine the data in columns C and D, that give the same values you have just
entered into B40 to B43 - . So far this is just revision of work you did in earlier
practicals.

Explain for someone new, why the formula SUM(D2:D37) which gives the
corrected ssq in cell D40 gives the same value as using the Excel function
DEVSQ in cell B40.
Explaining the variation
In the survey, there was also the information of which variety each farmer used.
The column is called Var for variety. It is a factor (or category) column with 3
leavels, namely NEW, OLD and TRAD. It is possible that some of the (currently
unknown) variability in the yields can be explained from the knowledge of the
variety of rice grown.
SADC Course in Statistics
Module I3 Sessions 6&7 – Page 1
Module I3 Sessions 6&7

Find how different the rice yields are for each variety. One way is to use SSCStat as follows:

Put the cursor within the data. Then use SSCstat => Analysis => Summary
Statistics The Variable to be analysed is y and the analysis is to be by a factor,
which is Var. Just calculate the means as shown in the table below.
Which statement below do you believe to be true?
The means are quite similar for the 3 varieties.
True/False
So I do not expect the knowledge of the variety to explain much of the
variation in the data.
The means are quite different for the 3 varieties.
True/False
So I expect the knowledge of the variety to explain quite a lot of the
variation in the data.
Column F in the Excel worksheet called ANOVA has copied the mean yields from the
summary you obtained. In column G the first row of data is 23.6, and this is calculated as
follows:
(45.4 – 40.6)² = 4.8² = 23.6
That is for variety OLD.

NEW
Explain how the other 2 values in column G are calculated. They correspond to
the variety NEW and TRAD are calculated:
(____ - 40.6)² = ___ = ____
TRAD
SADC Course in Statistics
Module I3 Sessions 6&7 – Page 2
Module I3 Sessions 6&7

Hence explain how the value 3528 is calculated – in cell G40. Also state how much
of the overall variation of the data is explained by the variety factor.
Calculated as follows:
Amount of variation
explained

Explain in words how the residual column (column I) is calculated and give
examples for the observations below.
The residual is the data value minus the ____________________
For the first observation it is therefore 53.6 – 45.4 = 8.2
For the 4th observation it is therefore 33.6 For the 5th observation
For the 8th observation

Explain the values in column J, including the corrected ssq in cell J40
Each value in J2 to J37 is the _______ of the values in ____. For example the value _____
in cell _____ is ______. These are therefore called the squared _________
Hence J40 is the residual ____________

We have chosen to calculate both columns G and J. How could we get the results
in cells G40 and J40 by just calculating one of them?
When you divide sums of squares by the corresponding degrees of freedom, you get mean
squares or variances. That is why this breakdown of the variability in the data is called the
Analysis of Variance, or ANOVA.
What you noted above – that the sums of squares add up – does not happen with the other
measures of variation, like the mean or the quartile deviations. The ONLY one of these
measures of variation that is “neat” mathematically, is the variance (and hence the standard
deviation).
That’s perhaps the main reason that this measure of variation is the most used and hence
the most important for you to understand.
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Module I3 Sessions 6&7 – Page 3
Module I3 Sessions 6&7

Take the sums of squares from row 40 and the degrees of freedom from row 41
and complete the table below:
Term
d.f.
ssq
msq
35
4955
142
ratio
Variety
Residual
Total
(A copy is at the bottom of the ANOVA_1 sheet. Check you have calculated this
correctly.)
The proportion of the variation explained by an explanatory measurement is called the
coefficient of determination and denoted by R². It is usually given as a percentage, so
multiply the result by 100.

Calculate R² for this case.
R² = _____%, i.e. _____ % of the variation in the yields can be explained by the different
varieties.
Finally we consider what this all means for the standard deviation. That is important
because standard deviations are easier to interpret than variances.
From the ANOVA table above, the overall standard deviation (see also cell B43) is s = √
142 = 11.9. Now after taking out the variation due to the varieties the unexplained
variation is s = √43 = 6.6. So the variance has been roughly divided by 4 and hence the
standard deviation has been roughly halved, by including the variety factor.
That result could be deduced from the value of R². It is roughly ¾, so ¼ of the variation
remains unexplained. The corresponding effect on the standard deviation is therefore to
halve it.
A second ANOVA table
The worksheet called ANOVA_2 in the same workbook has been started. It contains the
village information, rather than the varieties. It needs to be completed, in the same way as
the information on the worksheet for the varieties.
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Module I3 Sessions 6&7 – Page 4
Module I3 Sessions 6&7
It is slightly easier, because the villages are “in order” in the worksheet.

Drag down the 4 village means in column F, to complete the column. The original
values are in the Summary sheet if you make a mistake and need to start this
column again.

Then complete this sheet in the same way as the ANOVA_1 sheet. (Do as much
as possible without looking at the formulae in the other sheet.) Hence complete
the ANOVA table for the villages and give the coefficient of determination.
Term
d.f.
ssq
msq
35
4955
142
ratio
Village
Residual
Total
R² = _____%, i.e. _____ % of the variation in the yields can be explained by the different
villages.

Use the same logic as at the end of question 2 to calculate the residual standard
deviation, and explain how this could be deduced from the value of R².
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Module I3 Sessions 6&7 – Page 5
Module I3 Sessions 6&7
Reading CAST
A new section in CAST has been constructed for this SADC course to explain the same
ideas as above. It is section 3.4 and is called “Variation and Groups”. It would be good to
read this material in pairs, so you can discuss, and relate to the work in Excel, done above.

Read pages 3.4.1 and 3.4.2 to review the idea of explained and unexplained
variation.

In 3.4.2, for the Mooring’s data the overall standard deviation of the October to
December rainfall is about 100mm (s=109.2). Use the slider to give the seasonal
forecast that is about ¾ of the way from “Worthless” to “Good”. This makes the
unexplained standard deviation from the 3 forecasts around 50mm, or about half
the value without the forecast.
If the unexplained standard deviation is about halved, what roughly would to R² value be?
(Hint – see the end of question 2 above)
Explain briefly why having a forecast of the seasonal rainfall that has less than the overall
variation, would be a good thing. (Hint: The parallel with the example at the bottom of
page 3.4.2 may help)

Read page 3.4.3. Summarise the 2 ideas in this page, using either (or both) the
example of the varieties (question 2) or villages (question 3) above.
Variation between groups:
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Module I3 Sessions 6&7 – Page 6
Module I3 Sessions 6&7
Variation within groups:

Read page 3.4.4 and try the applet, making sure you click within the “Total”,
“Between” and “Within” boxes, to see which deviations are then displayed.

Also open the Excel workbook as before and use either the sheet ANOVA_1 or
ANOVA_2.
The text in CAST includes the following statement
The following relationship requires some algebra to prove but is important.

Which columns in the Excel sheet correspond to the different parts of this
formula? And which cells give the 3 values above of the ssq? Copy the appropriate
formula from above, into the table below:
Terms
Columns in Excel
Cell giving ssq
Formula
Between
Within
Total
 Move the slider on page 3.4.4 so about the same proportion of the variation is
explained as in the examples above for a) varieties and b) villages. What was the
residual (SSwithin ) in each case?
Same proportion as:
Residual ssq (SSwithin )
Varieties
Villages
 Read page 3.4.5. Try the 5 data sets and hence give a similar interpretation of the
R² values for the varieties and the villages.
Term
SADC Course in Statistics
Explanation
Module I3 Sessions 6&7 – Page 7
Module I3 Sessions 6&7
Varieties
Villages

Read page 3.4.6, the final page in this section.

Try the applet in the section titled “Illustration of calculations” with different
amounts of variation explained and different sample sizes. Try a number of
samples.

For a given position of the slider which mean squares stay roughly the same size
when the sample size is changed? Explain why that should be the case.
Roughly by how much does the remaining mean square change, as the sample size
changes? And what therefore happens to the ratio (called F in the applet)? What is the
practical importance of that result?
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Module I3 Sessions 6&7 – Page 8