Variability
Eamonn Mallon
26/08/2020
Descriptive statistics
- Measurements often cluster around intermediate values (central tendency)
- Variability of a sample is perhaps the most important quantity in data analysis
Variability
Variability is central to data analysis
- greater variability in data = greater uncertainity in the parameters calculated from this data
- => the lower our ability to distinguish between competing hypotheses
Measures of variability
- The x axis is just the order they were measured in, that is not important
- How can we quantify the variation (the scatter)?
The range
- The range is the difference between the lowest and highest values
- Too dependent on outlying values
- Ideally all data would contribute to our measure of variability
A slight detour: Residuals
- Calculate departures from the mean \( \bar{y}-y \)
- Add all them up \( \sum_{i=1}^{n} (\bar{y}-y_i) \)
- But that equals zero!!!!!
- Need to get rid of the signs
Behold! The sum of squares
- Ignore the signs, just use the absolute values
- This is exactly what newer techniques do, but the sums are hard
- So earlier techniques just squared the residuals before adding them
- \( \sum_{i=1}^{n} (\bar{y}-y_i)^2 \)
- The single most important quantity in all of statistics the sum of squares
- 102.5454545 units squared
But what happens if you add more data?
- The sum of squares gets bigger (unless your added data point was exactly the mean)
- Hopefully, everyone can see this isn't great
- Easily solved, just divide by n (mean squared deviation)
- But hang on, we need to detour to explain degrees of freedom
A slight detour: Degrees of freedom
- I have five numbers (positive or negative) that have to add up to 20
- What's the first number?
- Could be anything
- Lets say its 2
- What's the second number?
- Could be anything
- Lets say its 7
- What's the third number?
- Could be anything
- Lets say its 4
A slight detour: Degrees of freedom
- What's the four number?
- Could be anything
- Lets say its 0
- What's the fifth number number?
- It's got to be 7
- Degrees of freedom (d.f.)
- n - 1
- the sample size, n, minus the number of parameters, p, estimated from the data.
Variance
- \( \sum_{i=1}^{n} (\bar{y}-y_i)^2 \) we had to calculate the mean, that is we calculated one parameter
- so rather than divide by n we divide by n-1
- Mean squared deviation becomes \( \frac{\sum_{i=1}^{n} (\bar{y}-y_i)^2}{n-1} \)
- also known as the variance \( s^2 \)
- an unbiased estimate of the variance, because we have taken account of the fact that one parameter was estimated from the data prior to computation
- standard deviation is s, the square root of variance. We use this as s is in the same dimensions as the mean (e.g mean height = 150cm, variance = 36cm2 , standard deviation = 6cm)
Using Variance
Variance can be used in two main ways
- for establishing measures of unreliability
- for testing hypotheses (a future lecture)
A measure of unreliability
- As variance increases so would unreliability
- \( unreliability \propto s^2 \)
- Unreliability should go down as the number of samples increases
- \( unreliability \propto \frac{s^2}{n} \)
- Unreliability should be in the same dimensions as our measurements
- \( S.E._\bar{y} =\sqrt{\frac{s^2}{n}} \) Standard error of the mean (s.e.m)
- An example: 50 \( \pm \) 5 (mean \( \pm \) s.e.m)
A slight detour: Differences between populations and samples
- This made me scratch my head as an undergrad
- The population is, for example, all students in the room, the sample is the five people I picked out.
- the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean,
- whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean