AS statistics page 1
Statistics for AS Biology
Descriptive Statistics
mean
number of times each value
occurs
Repeated measurements in biology are
rarely identical, due to random errors
and natural variation. If enough
measurements are repeated they can be
plotted on a histogram, like the one on
the right. This usually shows a normal
distribution, with most of the repeats
close to some central value. Many
biological phenomena follow this
pattern: eg. peoples' heights, number of
peas in a pod, the breathing rate of
insects, etc.
95% CI
95% CI
normal
distribution
curve
values
The central value of the normal
distribution curve is the mean (also
known as the arithmetic mean or
average). But how reliable is this
mean? If the data are all close
together, then the mean is probably
good, but if they are scattered
widely, then the calculated mean
may not be very reliable. The width of the normal distribution curve is given by the standard
deviation (SD), and the larger the SD, the less reliable the data. For comparing different sets of data,
a better measure is the 95% confidence interval (CI). This is derived from the SD, and is the range
above and below the mean within which 95% of the repeated measurements lie (marked on the
histogram above). You can be pretty confident that the real mean lies somewhere in this range.
Whenever you calculate a mean you should also calculate a confidence limit to indicate the quality
of your data.
small confidence limit,
low variability,
data close together,
mean is reliable
large confidence limit,
high variability,
data scattered,
mean is unreliable
In Excel the mean is calculated using the formula =AVERAGE (range) , the SD is calculated using
=STDEV (range) , and the 95% CI is calculated using =CONFIDENCE (0.05, STDEV(range), COUNT(range)) .
This spreadsheet shows two sets of
data with the same mean. In group A
the confidence interval is small
compared to the mean, so the data are
reliable and you can be confident that
the real mean is close to your
calculated mean. But in group B the
confidence interval is large compared
to the mean, so the data are unreliable,
as the real mean could be quite far
away from your calculated mean. Note
that Excel will always return the
results of a calculation to about 8
decimal places of precision. This is meaningless, and cells with calculated results should always be
formatted to a more sensible precision (Format menu > Cells > Number tab > Number).
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AS statistics page 2
Plotting Data
Once you have collected data you will want to plot a graph or chart to show trends or relationships
clearly. With a little effort, Excel produces very nice charts. First enter the data you want to plot into
two columns (or rows) and select them.
Drawing the Graph. Click on the chart wizard
. This has four steps:
1. Graph Type. For a bar graph choose Column and for a scatter graph (also known as a line graph)
choose XY(Scatter) then press Next. Do not choose Line.
2. Source Data. If the sample graph looks OK, just hit Next. If it looks wrong you can correct it by
clicking on the Series tab, then the red arrow in the X Values box, then highlight the cells
containing the X data on the spreadsheet. Repeat for the Y Values box.
3. Chart Options. You can do these now or change them later, but you should at least enter suitable
titles for the graph and the axes and probably turn off the gridlines and legend.
4. Graph Location. Just hit Finish. This puts the chart beside the data so you can see both.
Changing the Graph. Once you have drawn the graph, you can now change any aspect of it by
double-clicking (or sometimes right-clicking) on the part you want to change. For example you can:
 move and re-shape the graph
 change the background colour (white is usually best!)
 change the shape and size of the markers (dots)
 change the axes scales and tick marks
 add a trend line or error bars (see below)
Lines. To draw a straight "line of best fit" right click on a point, select Add Trendline, and choose
linear. In the option tab you can force it to go through the origin if you think it should, and you can
even have it print the line equation if you are interested in the slope or intercept of the trend line. If
instead you want to "join the dots" (and you don't often) double-click on a point and set line to
automatic.
Error bars. These are used to show the confidence intervals on the graph. You must already have
entered the 95% confidence limits on the spreadsheet beside the X and Y data columns. Then
double-click on the points on the graph to get the Format Data Series dialog box and choose the Y
Error Bars tab. Click on the red arrow in the Custom + box, and highlight the range of cells
containing your confidence limits. Repeat for the Custom - box.
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AS statistics page 3
Problems
1.
Here are the results of an investigation into the rate of photosynthesis in the pond weed Elodea.
The number of bubbles given off in one minute was counted under different light intensities,
and each measurement was repeated 5 times. Use Excel to calculate the means and 95%
confidence limits of these results, then plot a graph of the mean results with error bars and a
line of best fit.
light
intensity
(Lux)
0
500
1000
2000
3500
5000
HGS Biology
repeat 1
repeat 2
repeat 3
repeat 4
repeat 5
5
12
7
42
45
65
2
4
20
25
40
54
0
5
18
31
36
72
2
8
14
14
50
58
1
7
24
38
28
36
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A2 statistics page 1
Statistics for A2 Biology
There is a bewildering variety of statistical tests available, and it is important to choose the right
one. This flow chart will help you to decide which statistical test to use, and the tests are described
in detail on the next 5 pages.
normal
data
Testing for a
correlation
non-normal Spearman correlation coefficient
data
=CORREL (range 1, range 2)
on ranks of data
0=no correlation/ 1=perfect correlation
Plot
scatter
graph
Finding how one
factor affects another
Testing for
a relation
between 2 sets
Calculate
mean and
95% CI from
replicates
Measurements
start
here
What
kind
of
test?
Testing for
a difference
between sets
Linear regression
Add Trendline to graph and
Display Equation.
Gives slope and intercept of line
same
individuals
Paired t-test
=TTEST(range1, range2, 2, 1)
If P<5% then significant difference
If P>5% then no significant difference
different
individuals
Unpaired t-test
=TTEST(range1, range2, 2, 2)
If P<5% then significant difference
If P>5% then no significant difference
2 sets
Plot
bar
graph
What
kind
of
data?
>2 sets
Frequencies (counts)
Comparing observed
counts to a theory
What
kind
of
test?
Testing for a difference
between counts
Testing for an association
between groups of counts
HGS Biology
Pearson correlation coefficient
=CORREL (range 1, range 2)
0 = no correlation
1 = perfect correlation
ANOVA
Tools menu > Data analysis > Anova
If P<5% then significant difference
If P>5% then no significant difference
2 test
=CHITEST(obs range, exp range)
If P<5% then disagree with theory
If P>5% then agree with theory
2 test
=CHITEST(obs range, exp range)
If P<5% then significant difference
If P>5% then no significant difference
2 test for association
=CHITEST(obs range, exp range)
If P<5% then significant association
If P>5% then no significant association
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A2 statistics page 2
Statistics to Test for a Correlation
Correlation statistics are used to investigate an association between two factors such as age and
height; weight and blood pressure; or smoking and lung cancer. After collecting as many pairs of
measurements as possible of the two factors, plot a scatter graph of one against the other. If both
factors increase together then there is a positive correlation, or if one factor decreases when the
other increases then there is a negative correlation. If the scatter graph has apparently random points
then there is no correlation.
variable 1
No Correlation
variable 2
Negative Correlation
variable 2
variable 2
Positive Correlation
variable 1
variable 1
There are two statistical tests to quantify a correlation: the Pearson correlation coefficient (r), and
Spearman's rank-order correlation coefficient (rs). These both vary from +1 (perfect correlation)
through 0 (no correlation) to –1 (perfect negative correlation). If your data are continuous and
normally-distributed use Pearson, otherwise use Spearman. In both cases the larger the absolute
value (positive or negative), the stronger, or more significant, the correlation. Values grater than 0.8
are very significant, values between 0.5 and 0.8 are probably significant, and values less than 0.5 are
probably insignificant.
In Excel the Pearson coefficient r is calculated using the formula: =CORREL (X range, Y range) . To
calculate the Spearman coefficient rs, first make two new columns showing the ranks (or order) of
the two sets of data, and then calculate the Pearson correlation on the rank data. The highest value is
given a rank of 1, the next highest a rank of 2 and so on. Equal values are given the same rank, but
the next rank should allow for this (e.g. if there are two values ranked 3, then the next value is
ranked 5).
In this example the size of breeding
pairs of penguins was measured to
see if there was correlation between
the sizes of the two sexes. The scatter
graph
and
both
correlation
coefficients clearly indicate a strong
positive correlation. In other words
large females do pair with large
males. Of course this doesn't say why,
but it shows there is a correlation to
investigate further.
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Linear Regression to Investigate a Causal Relationship.
If you know that one variable causes the changes in the other variable, then there is a causal
relationship. In this case you can use linear regression to investigate the relation in more detail.
Regression fits a straight line to the data, and gives the values of the slope and intercept of that line
(m and c in the equation y = mx + c).
The simplest way to do this in Excel is to
plot a scatter graph of the data and use the
trendline feature of the graph. Right-click on
a data point on the graph, select Add
Trendline, and choose Linear. Click on the
Options tab, and select Display equation on
chart. You can also choose to set the
intercept to be zero (or some other value).
The full equation with the slope and
intercept values are now shown on the chart.
In this example the absorption of a yeast cell suspension is plotted against its cell concentration
from a cell counter. The trendline intercept was fixed at zero (because 0 cells have 0 absorbance),
and the equation on the graph shows the slope of the regression line.
The regression line can be used to make quantitative predictions. For example, using the graph
above, we could predict that a cell concentration of 9 x 107 cells per cm3 would have an absorbance
of 1.37 (9 x 0.152).
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A2 statistics page 4
T-Test to Compare Two Sets of Data
Another common form of data analysis is to compare two sets of measurements to see if they are the
same or different. For example are plants treated with fertiliser taller than those without? If the
means of the two sets are very different, then it is easy to decide, but often the means are quite close
and it is difficult to judge whether the two sets are the same or are significantly different. To
compare two sets of data use the t-test, which tells you the probability (P) that there is no difference
between the two sets. This is called the null hypothesis.
P varies from 0 (impossible) to 1 (certain). The higher the probability, the more likely it is that the
two sets are the same, and that any differences are just due to random chance. The lower the
probability, the more likely it is that that the two sets are significantly different, and that any
differences are real. Where do you draw the line between these two conclusions? In biology the
critical probability is usually taken as 0.05 (or 5%). This may seem very low, but it reflects the facts
that biology experiments are expected to produce quite varied results. So if P > 5% then the two sets
are the same (i.e. accept the null hypothesis), and if P < 5% then the two sets are different (i.e. reject
the null hypothesis). For the t test to work, the number of repeats should be at least 5.
In Excel the t-test is performed using the formula: =TTEST (range1, range2, tails, type) . For the
examples you'll use in biology, tails is always 2 (for a "two-tailed" test), and type can be either 1 for
a paired test (where the two sets of data are from the same individuals), or 2 for an unpaired test
(where the sets are from different
individuals). The cell with the t test P
should be formatted as a percentage
(Format menu > cell > number tab >
percentage).
This
automatically
multiplies the value by 100 and adds the
% sign. This can make P values easier to
read and understand. It’s also a good idea
to plot the means as a bar chart with error
bars to show the difference graphically.
In the first example the yield of potatoes
in 10 plots treated with one fertiliser was
compared to that in 10 plots treated with
another fertiliser. Fertiliser B delivers a
larger mean yield, but the unpaired t-test
P shows that there is a 8% probability
that this difference is just due to chance.
Since this is >5% we accept the null
hypothesis that there is no significant
difference between the two fertilisers.
In the second example the pulse rate of 8
individuals was measured before and
after eating a large meal. The mean pulse
rate is certainly higher after eating, and
the paired t-test P shows that there is
only a tiny 0.005% probability that this
difference is due to chance, so the pulse
rate is significantly higher after a meal.
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A2 statistics page 5
ANOVA to Compare >2 sets of Data
The t test is limited to comparing two sets of data, so to compare many groups at once you need
analysis of variance (ANOVA). From the Excel Tools menu select Data Analysis then ANOVA
Single Factor. This brings up the ANOVA dialogue box, shown here.
 Enter the Input Range by clicking in
the box then selecting the range of
cells containing the data, including
the headings.
 Check that the columns/rows choice
is correct (this example is in three
columns), and click in Labels in
First Row if you have included
these. The column headings will
appear in the results table.
 Leave Alpha at 0.05 (for the usual
5% significance level).
 Click in the Output Range box and
click on a free cell on the worksheet,
which will become the top left cell
of the 8 x 15-cell results table.
 Finally press OK.
The output is a large
data table, and you may
need to adjust the
column widths to read
it all. At this point you
should plot a bar graph
using the averages
column for the bars and
the variance column for
the error bars.
The most important cell
in the table is the Pvalue, which as usual is
the probability that the
null hypothesis (that
there is no difference
between any of the data
sets) is true. This is the
same as a t-test
probability, and in fact if you try ANOVA with just two data sets, it returns the same P as a t test. If
P > 5% then there is no significant difference between any of the data sets (i.e. the null hypothesis is
true), but if P < 5% then at least one of the groups is significantly different from the others.
In the example on this page, which concerns the grain yield from three different varieties of wheat,
P is 0.14%, so is less than 5%, so there is a significant difference somewhere. The problem now is
to identify where the difference lies. This is done by examining the variance column in the summary
table. In this example, varieties 2 and 3 are very similar, but variety 1 is obviously the different one.
So the conclusion would be that variety 1 has a significantly lower yield than varieties 2 and 3.
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A2 statistics page 6
Chi-squared Test for Frequency Data
Sometimes the data from an experiment are not measurements but counts (or frequencies) of
things, such as counts of different phenotypes in a genetics cross, or counts of species in different
habitats. With frequency data you can’t usually calculate averages or do a t test, but instead you do a
chi-squared (2) test. This compares observed counts with some expected counts and tells you the
probability (P) that there is no difference between them. In Excel the 2 test is performed using the
formula: =CHITEST (observed range, expected range) . There are three different uses of the test
depending on how the expected data are calculated.
Sometimes the expected data can be calculated from a quantitative theory, in which case you
are testing whether your observed data agree with the theory. If P < 5% then the data do not
agree with the theory, and if P > 5% then the data do agree with the theory. A good example is a
genetic cross, where Mendel’s laws can be used to
predict frequencies of different phenotypes. In this
example Excel formulae are used to calculate the
expected values using a 3:1 ratio of the total
number of observations. The 2 P is 53%, which is
much greater than 5%, so the results do indeed
support Mendel’s law. Incidentally a very high P
(>80%) is suspicious, as it means that the results
are just too good to be true.
1.
2.
Other times the expected data are calculated by assuming that the counts in all the categories
should be the same, in which case you are testing whether there is a difference between the
counts. If P < 5% then the counts are significantly
different from each other, and if P > 5% then there
is no significant difference between the counts. In
the example above the sex of children born in a
hospital over a period of time is compared. The
expected values are calculated by assuming there
should be equal numbers of boys and girls, and the
2 P of 6.4% is greater than 5%, so there is no
significant difference between the sexes.
If the count data are for categories in two groups, then the expected data can be calculated by
assuming that the two groups are independent. If P < 5% then there is a significant association
between the two groups, and if P > 5% then the two groups are independent. Each group can have
counts in two or more categories, and the observed frequency data are set out in a table, called a
contingency table. A copy of this table is then made for the expected data, which are calculated for
each cell from the corresponding totals of the observed data, using the formula
E = column total x row total / grand total . In this example the flow rate of a stream (the two categories
fast / slow) is compared to the type of stream bed (the four categories weed-choked / some weeds /
shingle / silt) at 50
different sites to see if
there is an association
between them. The 2 P
of 1.1% is less than 5%,
so there is an association
between flow rate and
stream bed.
3.
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A2 statistics page 7
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A2 statistics page 8
Problems
1. In a test of two drugs 8 patients were given one drug and 8 patients another drug. The number of hours of relief from
symptoms was measured with the following results:
Drug A
3.2
1.6
5.7
2.8
5.5
1.2
6.1
2.9
Drug B
3.8
1.0
8.4
3.6
5.0
3.5
7.3
4.8
Find out which drug is better by calculating the mean and 95% confidence limit for each drug, then use an
appropriate statistical test to find if it is significantly better than the other drug.
2. In one of Mendel's dihybrid crosses, the following types and numbers of pea plants were recorded in the F2
generation:
Yellow round seeds
Yellow wrinkled seeds
Green round seeds
Green wrinkled seeds
289
122
96
39
According to theory these should be in the ratio of 9:3:3:1. Do these observed results agree with the expected ratio?
3. The areas of moss growing on the north and south sides of a group of trees were compared.
North side of tree
20
43
53
86
70
54
South side of tree
63
11
21
54
9
74
Is there a significant difference between the north and south sides?
4. Five mammal traps were placed in randomly-selected positions in a deciduous wood. The numbers of field mice
captured in each trap in one day were recorded. The results were:
Trap A
B
C
D
E
no. of mice 22
26
21
8
23
Trap D caught far fewer mice than the others. Did this happen by chance or is the result significant?
5. In an investigation into pollution in a stream, the concentration of nitrates was measured at six different sites, and a
diversity index was calculated for the species present.
Site
1
2
3
4
5
6
413.3
439.7
726
850
567.3
766.7
Conductivity (S)
Diversity index
7.51
5.17
4.49
3.82
5.88
3.74
Is there a correlation between conductivity and diversity, and how strong is it? (The diversity index is calculated
from biotic data, so is not normally distributed.)
6. The blood groups of 400 individuals, from 4 different ethnic groups were recorded with the following results:
Ethnic group
Blood Group O
Blood Group A
Blood Group B
Blood Group AB
1
46
40
7
3
2
48
39
12
2
3
53
33
12
4
4
55
30
13
3
Is there as association between blood group and ethnic group?
7. The effect of enzyme concentration on rate of a reaction was investigated with the following results.
Enzyme concentration (mM)
0
0.1
0.2
0.5
0.8
1.0
Rate (arbitrary units)
0
0.8
1.1
3.2
6.6
7.2
Plot a graph of these results, fit a straight line to the data, and find the slope of this line. Use the slope to predict the
rate at an enzyme concentration of 0.7mM.
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A2 statistics page 9
Alternative flow chart, showing the non-parametric tests (which are not available in Excel):
Parametric Test
Testing for a
correlation
Plot
scatter
graph
Finding how
one factor
affects another
Testing for
a relation
between 2 sets
Calculate
mean and
95% CI from
replicates
Pearson correlation coefficient
=CORREL (range 1, range 2)
0 = no correlation
1 = perfect correlation
same
individuals
Spearman correlation coefficient
=CORREL (range 1, range 2)
on ranks of data
0=no correlation
1=perfect correlation
Linear regression
Add Trendline to graph and
Display Equation.
Gives slope and intercept of line
Parametric Test
What
kind
of
test?
Non-Parametric Test
Paired t-test
=TTEST(range1, range2, 2, 1)
If P<5% then sig. difference
If P>5% then no sig. difference
Non-Parametric Test
Wilcoxon Matched Pairs test
Not available in Excel
2 sets
Measurements
Testing for
a difference
between sets
Parametric Test
different
individuals
Plot
bar
graph
Unpaired t-test
=TTEST(range1, range2, 2, 2)
If P<5% then sig. difference
If P>5% then no sig. difference
Parametric Test
start
here
What
kind
of
data?
>2 sets
Frequencies (counts)
Comparing observed
counts to a theory
What
kind
of
test?
Testing for a difference
between counts
Testing for an association
between groups of counts
ANOVA
Tools menu >Data analysis >Anova
If P<5% then sig. difference
If P>5% then no sig. difference
Non-Parametric Test
Mann-Whitney U-test
Not available in Excel
Non-Parametric Test
Kruskal-Wallis test
Not available in Excel
2 test
=CHITEST(obs range, exp range)
If P<5% then disagree with theory
If P>5% then agree with theory
2 test
=CHITEST(obs range, exp range)
If P<5% then sig. difference
If P>5% then no sig. difference
2 test for association
=CHITEST(obs range, exp range)
If P<5% then sig. association
If P>5% then no sig. association
http://web.me.com/mindorffd/Site/Graph_plotting_files/StatisticsWorksheets.doc
HGS Biology
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