1. POPULATION DYNAMICS Required background knowledge:
Data and variability concepts
Data collection
Measures of central tendency (mean, median, mode, variance, stdev)
Normal distribution and SE
Student’s t-test and 95% confidence intervals
Chi-Square tests
MS Excel

2. MEAN (average) MEAN (average) MODE MEDIAN
STATISTICS: DESCRIBING VARIABILITY
Measures of Central Tendency
MEAN (average)
MEAN (average)
MODE
MEDIAN

3. X Measures of Central Tendency MEAN (average)
STATISTICS: DESCRIBING VARIABILITY: MEANS
Measures of Central Tendency X
MEAN (average)
Arithmetic mean or Average
N = 106
Σ = 182.4
Mean = 1.72
Mean = Σx N
Population mean = μ sample mean = x
We use x as a proxy for μ

4. MEAN (average) MEAN (average) MODE MEDIAN
STATISTICS: DESCRIBING VARIABILITY
Measures of Central Tendency
MEAN (average)
MEAN (average)
MODE
MEDIAN

5. The most commonly represented X value
STATISTICS: DESCRIBING VARIABILITY: MODE
MODE:
The most commonly represented X value

6. MEAN (average) MEAN (average) MODE MEDIAN
STATISTICS: DESCRIBING VARIABILITY
Measures of Central Tendency
MEAN (average)
MEAN (average)
MODE
MEDIAN

7. Median – the middle value in a ranked data set
STATISTICS: DESCRIBING VARIABILITY: MEDIAN
Median – the middle value in a ranked data set

8. If there are an ODD number of data points this is easy
STATISTICS: DESCRIBING VARIABILITY: MEDIAN
Step 1 – Order the data from low to high
Step 2 – Determine the middle data point
If there are an EVEN number of data points you will need to interpolate
If there are an ODD number of data points this is easy
The middle data point lies half way between that associated with observation no 5 (1.75) and observation no 6 (1.8) = 1.775
Can be calculated as either:
a.) ( ) / 2
b.) ((1.8 – 1.75) / 2)

9. MEDIAN MODE MEAN (average)
STATISTICS: DESCRIBING VARIABILITY
MEDIAN
MODE
MEAN (average)
Measures of Central Tendency
FREQUENCY TABLES

10. The most commonly represented X value
STATISTICS: DESCRIBING VARIABILITY: MODE
Modes and FREQUENCY TABLES
MODE
The most commonly represented X value
How? Construct a frequency table from the data: whichever “class” of data occurs at the highest frequency is the MODE
Classes should be calculated in EVEN intervals from smallest to largest value of x

11. STATISTICS: DESCRIBING VARIABILITY: MODE
UNIMODAL
BIMODAL
TRIMODAL

12. MEDIAN MODE MEAN (average)
STATISTICS: DESCRIBING VARIABILITY
MEDIAN
MODE
MEAN (average)
Measures of Central Tendency
FREQUENCY TABLES

13. EXCEL EXERCISES Open UWC Excel Exercises.xls
Activate ‘Length’ worksheet
Calculate the mean =average(data:range)
Calculate the mode =mode(data:range)
Calculate the median =median(data:range)
Activate ‘Height’ worksheet
Construct a frequency table
Type in specified class intervals (bin range)
Calculate the frequency of values for each class interval
TOOLS MENU < DATA ANALYSIS < HISTOGRAM
Point Excel to Input Range (original dataset)
Point Excel to the Bin Range
Point Excel to an Output Range (any empty cell)
Click Ok
Construct a graphs of the values in the output range
Use the Bin Range as labels for your graph

14. Measures of Dispersion
STATISTICS: DESCRIBING VARIABILITY
Measures of Dispersion
Measures of Central Tendency
MEDIAN
MODE
MEAN (average)
FREQUENCY TABLES
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION
how data are distributed around the mean
Average, middle or most common data point
MEAN
MEAN

15. Measures of Dispersion
STATISTICS: DESCRIBING VARIABILITY
Measures of Dispersion
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION

16. Range: STATISTICS: MEASURES OF DISPERSION: RANGE
Essentially the lowest and highest value in the data set
N.B. Subject to measurement errors, typographic mistakes and outliers

17. Measures of Dispersion
STATISTICS: DESCRIBING VARIABILITY
Measures of Dispersion
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION

18. 1/4 1/4 1/4 1/4 Inter-Quartile Range Inter-Quartile Range:
STATISTICS: MEASURES OF DISPERSION: INTERQUARTILE RANGE
Inter-Quartile Range:
In a ranked data set, those values corresponding to ¼ (lower or 25% quartile) and ¾ (upper or 75% quartile) of the observations: 50% of the observations lie between these two values
MEAN
¼ of the way through this ranked data set of 9 values = observation number 2.25 (=9 x 0.25)
Calculate the data point that would be associated with observation number 2.25 by interpolation between observation numbers 2 (1.45) and 3 (1.6) i.e. = ((1.6 – 1.45) * 0.25)
= ((0.15) * 0.25) =
= (Lower Quartile)
1/4
1/4
1/4
1/4
DITTO for 75% Quartile………………..
25% Quartile
75% Quartile
Inter-Quartile Range
To give us an interquartile range: –
If you have to use a range, use the inter-quartile range as it ignores outliers

19. Measures of Dispersion
STATISTICS: DESCRIBING VARIABILITY
Measures of Dispersion
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION

20. STATISTICS: MEASURES OF DISPERSION: CUMULATIVE FREQUENCY
Can also calculate Cumulative Frequency
THEN
Draw a Graph of Cumulative Frequency (Y) against Ordered Data (X) on an X-Y PLOT
THEN
Calculate Lower and Upper Quartiles from Figure
UPPER
MEDIAN
LOWER

21. Measures of Dispersion
STATISTICS: DESCRIBING VARIABILITY
Measures of Dispersion
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION

22. =(x-mean) =(x-mean)*-1 =average(!x-mean!)
STATISTICS: MEASURES OF DISPERSION: MEAN DEVIATION
Subtract the Mean from each x value
=(x-mean)
Convert negatives to positives to give overall deviation from the mean
=(x-mean)*-1 Σ
SUM, Divide by N to give average deviation of any data point from the mean – MEAN DEVIATION
=average(!x-mean!) N
mean

23. Measures of Dispersion
STATISTICS: DESCRIBING VARIABILITY
Measures of Dispersion
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION

24. Sum of Squares (Sample) Mean Sum of Squares (sample)
Variance and Standard Deviation
There is another way to remove the negatives – and that is to square the (x – mean) values
Always = Zero Σ N
mean
Length (mm) of Drosophila melanogaster Instar III larvae
=(x-mean)^2 Σ N
mean Σ N
mean
Sum of Squares (Sample)
Mean Sum of Squares (sample)
(VARIANCE)
(Variance) = √
Standard Deviation (sample)
s = 1.5
=SQRT(variance)
The values of x = 4, sample variance (2.25) and sample standard deviation (1.5) ALL refer to the sample of 16 measures 16
2.25

25. The smaller the standard deviation, the closer the data are to the mean
The bigger the standard deviation, the greater the spread of data around the mean – the greater the variability
Mean s2 N s

26. EXCEL EXERCISES 2 Open UWC Excel Exercises.xls
Activate ‘Dispersion measures’ worksheet
Calculate the maximum =max(data:range)
Calculate the minimum =min(data:range)
Calculate the range =max-min
Calculate the inter-quartile range using formulas
Copy the original data to the Ordered Data column
Sort in ascending order
Enter Order No. For each value (use the autofill handle)
Calculate the lower (25%) quartile order number and value (use the formulas provided)
Calculate the upper (75%) quartile order number and value (use the formulas provided)
Interquartile range = Upper quartile value – lower quartile value
Calculate the inter-quartile range using XY scatterplot
Calculate the Mean Deviation =var(data:range)
Calculate the Standard deviation =stdev(data:range)

27. POPULATION statistics
Samples as estimators for populations
POPULATION statistics
SAMPLE statistics μ x
PROXY
Mean σ2 s2
Variance
PROXY σ s
Standard Deviation
SS denominator
n-1 n X 5
Mean N 25
Total 7 6 4 3
WHY? Because not all the measures are completely independent of each other.
In this table of five measures, the total is 25 and x is 5 6 5 4 3
Collected only the first four then total = 18
So…last value would HAVE TO BE 7 to get mean of 5
So…last number is NOT independent of the others
For populations data MUST be independent data.
So…when we calculate σ2 we divide the sum of squares by (n-1) and NOT n (as previously): σ is still calculated as
(n-1) = Degrees of Freedom = v √ σ2

28. Variability = Uncertainty
STATISTICS: DESCRIBING VARIABILITY
Variability = Uncertainty
Probabilities
Measures of Dispersion
CUMULATIVE FREQUENCIES
INTER-QUARTILE RANGE
RANGE
MEAN DEVIATION
VARIANCE and
STANDARD DEVIATION