1. 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
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2. 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
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3. Measures of Spread Range Inter Quartile Range Variance
Standard Deviation
This hour lesson we concentrate on how to calculate the following two measures
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4. Range = largest value - smallest value
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5. 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

6. Interquartile Range = Q3 - Q1 = 525 - 445 = 80
L25= (n+1)*25/ /4 = th value
L75= (n+1)*75/ *3/4 = th value
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = = 80
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7. Basic Notation
As we will be working with formulas we need to make sure about some notation
Data set “X”
10, 30, , , , 21, , , 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
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8. Net deviations from the mean will always sum to zero
So “total distance” from the mean is zero
Because +ve and –ve contributions cancel
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9. 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
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10. 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
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11. Population Variance for Grouped Data
Mi is calls midpoint our Xi
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12. 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!
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13. Formulae RAW DATA Sample Variance RAW DATA Population Variance
GROUPED DATA
Sample Variance
GROUPED DATA
Population Variance
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14. 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!!
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15. 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
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16. 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
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17. Taking Lots of Samples of fixed size n & Build distributions of S2 and σ2
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18. 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
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19. 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
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20. 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
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