1. Central tendency and spread
Stats Club 7
Marnie Brennan

2. References
Petrie and Sabin - Medical Statistics at a Glance: Chapter 5, 6, 10, 35 Good
Petrie and Watson - Statistics for Veterinary and Animal Science: Chapter 2, 4 Good
Thrusfield – Veterinary Epidemiology: Chapter 12
Kirkwood and Sterne – Essential Medical Statistics: Chapter 4

3. Terminology!
Along similar lines of previous Stats Clubs, we are talking about ways of describing your numerical data
Gives you basic calculations to do to explore your data (get a feel for it)
Enables you to compare your data with those collected by other researchers

4. Central tendency
Central tendency = a measure of location or position of data, i.e. the ‘average’
This basically means calculating things like:
Mean (arithmetic mean)
Median
Mode
Others
E.g. geometric mean (distn. skewed to the right), weighted mean
Nice table in Petrie and Sabin (Chapter 5) summarising advantages and disadvantages of all measurements

5. Central tendency – Mean, Median
Mean = Sum of your data/total number of measurements
Algebraically defined
Affected by skewed data THEREFORE good to use for normally distributed variables
Median = The midpoint of your values i.e. what the ‘halfway’ value in your data is
If the observations are arranged in increasing order, the median would be the middle value
Not algebraically defined
Not affected by skewed data THEREFORE good to use for non-normally distributed variables

6. Distributions
Median
Mean
Mean and median the same

7. Central tendency - Mode
Mode = the value that occurs the most frequently in a data set
Generally means more if you have discrete data e.g. The most common litter size of bearded collie dogs is 7
Not often used
What is the mode?

8. Spread
Spread = measure of dispersion or variability (variation) of data
This basically means calculating things like:
Range
Percentiles (Quartiles, Interquartile range)
Variance
Standard deviation
Others
E.g. coefficient of variation
Nice table in Petrie and Sabin (Chapter 6) summarising main points about these measurements

9. Range and percentiles
Range = the range between the minimum and maximum values of your data
Gives an indication of spread at a very basic level
Distorted by outliers (get a large range)
Percentiles = if data is ordered from lowest to highest, these divide the data up into ‘compartments’
E.g. The 5th percentile = the point along the data below which 5% of the data lies; the 20th percentile = the point in the data below which 20% of the data lies
Special types of percentiles are called ‘quartiles’ – these divide the data into 4 equal parts (the 25th, 50th and 75th percentiles)
From these, you get an ‘interquartile range’ - IQR, which is values between the 25th and 75th percentiles
The 50th percentile is the median
Not distorted by outliers

10. Range = (6)
Q1 (25th percentile) = 24
Q3 (75th percentile) = 26
IQR = (2)
Range = (133.9)
Q1 (25th percentile) = 6
Q3 (75th percentile) = 36
IQR = 6-36 (30)
What conclusions can we draw about what to use when??

11. Rule of thumb
Mean and range = good to use for normally distributed variables
Median and interquartile range = good to use for non-normally distributed variables

12. Variance Variance = the deviations of the data values from the mean
e.g. If the data are bunched around the mean, the variance is small; if the data are spread out, the variance is large
Calculated by squaring each distance between the observations and the mean - we then take the mean of this (add all values together and divide by the total number of observations minus 1)
Reason for squaring – the negative values are ‘cancelled’ out
DON’T WORRY ABOUT HOW TO DO THIS! This is what computers are for!
Measured in the same units as the observations, but squared e.g. If the units are grams, the variance will be in grams squared

13. Mean = 26
Variance = 430
Mean = 23
Variance = 11090

14. Example If we had 6 observations (with mean = 0.17):
15, 18, -14, -17, -3 and 2
What is the variance?
= (15 – 0.17)2 + ( ) 2 + (-14 – 0.17) 2 + (-17 – 0.17) 2 + (-3 – 0.17) 2 + (2-0.17) 2/6-1 =
It is n-1 to reduce bias (we have a sample vs. whole population - again don’t worry too much!)

15. Standard Deviation (SD)
Standard deviation = square root of the variance
Similar to variance – also relates to the deviations of the observations from the mean (the ‘average’ deviation)
Therefore the units are the same as for the observations – more convenient than variance?
If we have a normally distributed dataset, then the mean +/- 2 x standard deviations approximately encompasses the central 95% of observations

16. Mean = 26
Variance = 430
Standard deviation = 21
Mean = 23
Variance = 11090
Standard deviation = 105

18. What about the standard error of the mean (SE or SEM)?
Relates to the precision of the sample mean as an estimate of the population mean
Standard deviation describes the variation in the data values from the mean
Can use SEM to construct confidence intervals
This will be covered in greater detail in another session

19. General rule
Standard deviation, variance and SEM are for normally distributed variables
For non-normally distributed variables, stick with interquartile range

20. Examples in papers

21. = Equal variances? Comparing groups of numerical values
It is an assumption of some of the tests used to compare different numerical data groups (e.g. T-tests, ANOVAs) that the variances must be equal (homogeneity of variance) in the groups compared
You need to know whether your groups meet these criteria – if they do not:
Use other non-parametric tests
Transform your data to fit the assumptions
Again DON’T WORRY – we’ll cover this in another session

22. When we start again… The bunfight that is: P-values.................!
Type I and Type II errors