Recap All about measures of location Mean Median Mode measures of centre Mean Median Mode measures of Any Position Percentiles You should be able to calculate these from grouped and raw data You should also be able to draw a box and whisker plot MH-Variance -Kuwait

This week Measures of Spread Sample of Heights of peoples in Coventry and Norwich We need more then the mean to compare data sets We need a numerical measure representing how the data varies MH-Variance -Kuwait

Measures of Spread Range Inter Quartile Range Variance Standard Deviation This hour lesson we concentrate on how to calculate the following two measures MH-Variance -Kuwait

Range = largest value - smallest value MH-Variance -Kuwait

Interquartile Range The interquartile range of a data set is the difference between the third quartile and the first quartile. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. 375 400 425 450 475 500 525 550 575 600 625

Interquartile Range = Q3 - Q1 = 525 - 445 = 80 L25= (n+1)*25/100 71/4 = 17.75 18th value L75= (n+1)*75/100 71*3/4 = 53.25 53th value 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80 MH-Variance -Kuwait

Basic Notation As we will be working with formulas we need to make sure about some notation Data set “X” 10, 30, 301 , 46, 18, 21, 19, 83, 4, .............., 88 x1 x2 x3 x4 x5 x6 x5 x6 x7 xn We often refer to a data set with an upper case letter like X, In which case the numbers in the data set are called elements (x1, x2, ..., xn) “n” or “N” is the number of elements or observations MH-Variance -Kuwait

Net deviations from the mean will always sum to zero So “total distance” from the mean is zero Because +ve and –ve contributions cancel MH-Variance -Kuwait

Measures of data Spread But we want a measure that will represent these net deviations somehow. One way to ensure a non-zero result is to square each deviation before adding it. We can then average these deviations by dividing by their number “n” and use this compare data sets OR, we can average and take the square root of the above This latter approach will have the same units as the underlying data. Variance Units squared Standard deviation Units of Units MH-Variance -Kuwait

Calculate the Variance for the following data set This data relates to Measures of distance travelled to work in units of (miles) 10 3.5 27 12 2 -0.9 -7.4 16.1 1.1 -8.9 0.81 54.76 259.21 1.21 79.21 395.2 This is the population variance (miles2) This is the population standard deviation (miles) Mean is 10.9 n=5 Units in miles MH-Variance -Kuwait

Population Variance for Grouped Data Mi is calls midpoint our Xi MH-Variance -Kuwait

Variance for Grouped Data For sample data For population data Sample variance s2 is commonly referred to by σ2n-1 Sample Standard Deviation s is commonly referred to by σn-1 So why is the sample measure divided by (n-1) ? – will deal with this soon! MH-Variance -Kuwait

Formulae RAW DATA Sample Variance RAW DATA Population Variance GROUPED DATA Sample Variance GROUPED DATA Population Variance MH-Variance -Kuwait

Things will now do 1- Understand why the following two formulas are the same and appreciate that the second form is much quicker to calculate than the first form 2- I would like you think of calculating variance as or Where Sxx can be calculated in different ways and can be divided appropriately dependent on whether we have a sample or population 3- We should investigate why we average , S2 , by (n-1) when we are dealing with a sample We will deal with this third and unusual point next!! MH-Variance -Kuwait

Why we divide by (n-1) Population Sample We take a random sample from the population and use it to estimate σ2 v v Sample v v MH-Variance -Kuwait

We are trying to estimate the true population mean σ2 In the real world we take a sample and use it Population Sample I am going to show you that S2 will be the better estimator of the true population variance, σ2 MH-Variance -Kuwait

Taking Lots of Samples of fixed size n & Build distributions of S2 and σ2 MH-Variance -Kuwait

Calculating s2 and σ2 of many samples , grouping and counting we can build distributions for S2 dist’n σs2 dist’n <σ2 σ2 RED distribution is centered around the real population variance MH-Variance -Kuwait

Showing = σ2 I will generate a Population of numbers AVG(S2) AVG(σs2) Row 1 Sample 1 S2 σs2 Row 2- Sample 2 S2 σs2 I will generate a Population of numbers And calculate the Pop Var (σ2) Row 3 Sample 3 S2 σs2 Row 4 Sample 4 S2 σS2 Then show that AVG(S2) = σ2 AVG(σs2) < σ2 Therefore E(S2)= σ2 Row 100 Sample 100 S2 σs2 MH-Variance -Kuwait

Summary Some questions We have looked at the formula for calculating Variance and Its square root Std- Deviation We have noted that we average by n or n-1 depending on whether or not we are working with a sample or population We have noted that that we can write Sxx = in different ways that are faster to calculate. We should work these different ways through shortly But first Some questions MH-Variance -Kuwait