1. Spread of Data
Whilst we have been looking at averages: mean, medians and modes these tell us little without knowing how the data spreads out around these central measures.
The quartiles mentioned in the last section can be used to describe how data spreads around the median. They give us the range of the middle 50% of the data. This is also known as the Interquartile Range and is often represented in a Boxplot.
The variance and the standard deviation are used in a similar way to describe how the data spreads around the mean. These will be used frequently in topics throughout S1 and S2. The standard deviation is the square root of the variance.

2. The following data is the ages of children in a play school
1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,5
Find the mean average age.
Now calculate how much each childs age differs from the mean.
How would you program a robot to calculate this?
How would you differentiate between above and below the mean?
If we want to know how their ages spread around the mean what is wrong with just adding these differences together and dividing by 16?

3. How the data varies about the mean average is known as the variance.
We can calculate it for lists of data by comparing each result to the mean.
Subtract the mean from each piece of data and square it (to make it positive so they don't cancel each other out)
Add them together and divide by how many pieces of data there are.

4. The data above can be put into a frequency table
1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,5
The data above can be put into a frequency table
How might we find the variance using this table rather than finding the difference between each piece of data?

5. This isn't the quickest method although it does make the most sense
The formulae can be manipulated into this form which is a lot easier to calculate
Because we have squared x the units x is measured in have also been squared. This poses a problem when interpreting the spread of data so they square root the variance to give the standard deviation.

6. Once you've matched them up decide the black and red sets of data are examples of what?
Weights of the 10 babies born in Arrowe Park hospital on 3/11/06
Heights of year 11 students at St Anselms College
Size of houses in Bebington
Children who attend churches on the Wirral
Students in UK
Babies born in November 2006
Number of bedrooms in the houses in Oaklands Drive
Age of 12 children in St Andrews Sunday School

7. To find the sample mean is exactly the same as the population mean
To find the sample mean is exactly the same as the population mean. However the sample variance must be calculated differently.
Compare the differences and similarities between this formula and the one for populations.
Remember we use
for populations and
for samples

8. Page 40
Exercise 3C
Q3, 4 and 6
Page 43
Exercise 3D
Q5 and 6
Page 49
Mixed Exercise 3F

9. Which average would be the best way to represent this continuous data?
1,1,1,1,1,2,2,3,3,3,4,4,6,11,22
Why?
Group this data with class widths: 1-2, 3-4, 5-7, 8-12, 13-22
Now sketch a histogram for it.
example of skewed data.agg

10. This data is an example of positive skew
Although it can be seen in this example other data sets may be only slightly skewed and less noticible. It is also a long winded way to decide on skew - drawing a histogram. There are other methods which involve the data we have been learning to calculate on this course.
Work out Q1,Q2 and Q3 for this data.
By calculating the differences between the median and the two quartiles decide which side of the boxplot is bigger? (Left or right)
We can use this technique to measure negative and psitive skew.

11. Other measures for skewnwss include:
positive skew: mode < median < mean
negative skew: mean < median < mode or
if we want to quantify the skewness as well as identify its direction we may use:
3(mean - median)
standard deviation
the closer the value to zero teh nearer a symmetrical distribution it is - normal

12. The Coeffient of Variation, V, is defined as: V = 100 x
There are other ways to interpret measures of location (averages) and spread (IQR and s.d.)
Often these values are of most use when comparing two or more data sets rather than singular use.
The Coeffient of Variation, V, is defined as:
V = 100 x
This gives a percentage of dispersion.
The Quartile coefficient of variation, QV, is:
QV = 100 x 0.5(Q3 - Q1) Q2
This last one being more useful when you have outliers you feel are distorting your data and you wish to ignore them.

13. Again there are many ways people use but one Edexcel use is:
Lastly we should add some ways to quantify outliers which continue to be mentioned:
Again there are many ways people use but one Edexcel use is:
if x -
is less greater than 2 or less than -2 then that value of x is considered an outlier

14. Page 71
Exercise 4F
Q1 - 4
Page 72
Mixed Exercise 4G
Q1, 3, 4, 7 and 8