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

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

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

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

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

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

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

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.) (1.75 + 1.8) / 2 b.) ((1.8 – 1.75) / 2) + 1.75

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

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

STATISTICS: DESCRIBING VARIABILITY: MODE UNIMODAL BIMODAL TRIMODAL

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

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

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

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

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

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

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) + 1.45 = ((0.15) * 0.25) + 1.45 = 0.0375 + 1.45 = 1.4875 (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: 1.4875 – 1.8875 If you have to use a range, use the inter-quartile range as it ignores outliers

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

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

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

=(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

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

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

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

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)

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

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