Statistics

The usual course of events for conducting scientific work “The Scientific Method” Reformulate or extend hypothesis Develop a Working Hypothesis Observation Conduct an experiment or a series of controlled systematic observations Appropriate statistical tests Confirm or reject hypothesis

The usual course of events for conducting scientific work “The Scientific Method” Reformulate or extend hypothesis Develop a Working Hypothesis Observation Conduct an experiment or a series of controlled systematic observations Appropriate statistical tests Confirm or reject hypothesis In a group of crickets, small ones seem to avoid large ones There will be movement away from large cricket by small ones Record the number of times that small crickets move away from small and large crickets. Chi square test There is a significant difference in the number of times small crickets move away from large vs. small ones Avoidance may depend on previous experience

Imagine that you are collecting samples (i.e. a number of individuals) from a population of little ball creatures - Critterus sphericales Little ball creatures come in 3 sizes: Small = Medium = Large =

-sample 1 -sample 2 -sample 3 -sample 4 -sample 5 You end up with a total of five samples

The real population (all the little ball creatures that exist) Your samples

Each sample is a representation of the population BUT No single sample can be expected to accurately represent the whole population

To be statistically valid, each sample must be: 1) Random: Thrown quadrat?? Guppies netted from an aquarium?

Assign numbers from a random number table

To be statistically valid, each sample must be: 2) Replicated:

But not - ‘Pseudoreplication’ Not pseudoreplication Pseudoreplication 10 samples from the same tree 10 samples from 10 different trees Sample size = 1Sample size = 10

TYPES OF DATA

RATIO DATA - constant size interval - a zero point with some reality e.g. Heights, rates, time, volumes, weights

INTERVAL DATA - constant size interval - no true zero point zero point depends on the scale used e.g. Temperature

Ordinal Scale - ranked data -grades, preference surveys

Nominal Scale Team numbers Drosophila eye colour

The kind of data you are dealing with is one determining factor in the kind of statistical test you will use.

Statistics and Parameters

Measures of: Central tendency - mean, median, mode Dispersion - range, mean deviation, variance, standard deviation, coefficient of variation

The real population (all the little ball creatures that exist) Central tendency - Mean

The real population (all the little ball creatures that exist) Your samples

The real population (all the little ball creatures that exist)  =  X i N Central Tendency 1) Arithmetic mean At Population level Measuring the diameters of all the little ball creatures that exist  population mean X i - every measurement in the population N - population size

Your samples X =  X i n n n n n

n Sample mean Sum of all measurements in the sample Sample size

If you have sampled in an unbiased fashion X =  X i n n n n n Each roughly equals 

Central tendency - Median Median - middle value of a population or sample e.g. Lengths of Mayfly (Ephemeroptera) nymphs 5 th value (middle of 9)

Median value Odd number of valuesEven number of values Median = middle value Median = + 2

Odd number of values (i.e. n is odd) Even number of values Or - to put it more formally Median = X (n+1) 2 Median = X (n/2) + X (n/2) + 1 2

Frequency (= number of times each measurement appears in the population Values (= measurements taken) c. Mode - the most frequently occurring measurement Mode Central tendency - Mode

Measures of Dispersion Why worry about this?? -because not all populations are created equal Distribution of values in the populations are clearly different BUT means and medians are the same Mean & median

Measures of Dispersion - 1. Range - difference between the highest and lowest values Remember little ball creatures and the five samples Range = -

Range - crude measure of dispersion Note - three samples do not include the highest value and - two samples do not include the lowest

Measures of Dispersion - 2. Mean Deviation X is a measure of central tendency Take difference between each measure and the mean X i - X BUT  X i - X = 0 So this is not useful as it stands

Measures of Dispersion - 2. Mean Deviation (cont’d) But if you take the absolute value -get a measure of disperson  |X i - X| and n = mean deviation

Measures of Dispersion - 3. Variance -eliminate the sign from deviation from mean Square the difference (X i - X) 2 And if you add up the squared differences - get the “sum of squares”  (X i - X) 2 (hint: you’ll be seeing this a lot!)

Measures of Dispersion - 3. Variance (cont’d) Sum of squares can be considered at both the population and sample level ss =  (X i - X) 2 SamplePopulation SS =  (X i -  ) 2

s 2 =  (X i - X) 2 SamplePopulation  2 =  (X i -  ) 2 Measures of Dispersion - 3. Variance (cont’d) If you divide by the population or sample size - get the mean squared deviation or VARIANCE N n-1 Population variance Sample variance

s 2 =  (X i - X) 2 Note something about the sample variance n-1 Measures of Dispersion - 3. Variance (cont’d) Degrees of freedom or df or

Measures of Dispersion - 4. Standard Deviation - just the square root of the variance  =  (X i -  ) 2 N Population Sample s =  (X i - X) 2 n-1

Standard Deviation - very useful Most data in any population are within one standard deviation of the mean

NORMAL DISTRIBUTION

Type of data Discrete Continuous Other distributions 2 categories & Bernoulli process > 2 categories Use a Binomial model to calculate expected frequencies Use a Poisson distribution to calculate expected frequencies From previous slide show

Type of data Discrete Continuous Other distributions 2 categories & Bernoulli process > 2 categories Use a Binomial model to calculate expected frequencies Use a Poisson distribution to calculate expected frequencies Now we’re dealing with:

Normal Distribution - bell curve

Central Limit Theorem Any continuous variable influenced by numerous random factors will show a normal distribution.

Normal curve is used for: 2) Continuous random data Weight, blood pressure weight, length, area, rates Data points that would be affected by a large number of random (=unpredictable) events Blood pressure age physical activity smokingdiet genes stress

Normal curves can come in different shapes So, for comparison between them, we need to standardize their presentation in some way

Standarize by calculating a Z-Score Z = value of a random variable - mean standard deviation Z = X - µ  or

Example of a z-score calculation The mean grade on the Biometrics midterm is 78.4 and the standard deviation is 6.8. You got a 59.7 on the exam. What is your z-score? Z = X - µ  Z = =

If you look at the formula for z-scores: z = value of a random variable - mean standard deviation z is also the number of standard deviations a value is from the mean

Each standard deviation away from the mean defines a certain area of the normal curve