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Measures of variation

Standard Deviation - A measure of how spread out numbers are, units are the same as the original data

Variance - The average of the squared difference from the mean

Range - Difference between largest and smallest

Interquartile range - Upper quartile - lower quartile the spread of the middle 50% of the data

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Line of best fit

When a linear relationship exists between two variables, a line of best fit can be drawn

It should always pass through the mean point of the graph

It can then be used to determine values when another is known

Regression Lines

Another name for a line of best fit calculated mathematically to ensure it produces the most accurate result

Relationship between the correlation coefficient and description of the data
r≈ 1 = Strong positive linear correlation
r≈ 0  = No linear correlation
r≈ -1 = Strong Negative linear correlation
When r = + or - 1, it is perfect correlation; the data lies in a perfectly straight line
When r=0, there is no linear relationship between the variable

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A scatter graph is a useful tool when determining whether a relationship exists between 2 variables

It’s usual to draw the independant variable on the horizontal axis and the dependant variable on the vertical axis

Correlation

A measure of relationship- the stronger the correlation, the more closely related the variables are likely to be

The type of correlation can be described in terms of the gradient of he slope formed by the points and how close the points lie to the straight line

Correlation Causation

Causation - The act or process of causing something

Even if there is a correlation within the data, it doesn’t mean that there. Is a direct variable. It could be caused by a third variable

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Used to calculate an estimate of the median for a grouped set of data

It can also be used to calculate the interquartile range

Cumulative frequency is shown along the vertical axis

Looks like:

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Skewness - Measures the distribution
Positive Skew - Median is closer to lower quartile
Negative Skew - Median is closer to upper quartile
Normal Distribution - Median is near the middle

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Some sets of data can be summarised using 5 key points:

1. The median value (Q2) - the middle number once the data has been written in order

2. Lower Quartile (Q1) - The value which lies one quarter of the way through the ordered data

3. Upper Quartile (Q3) - The value which lies three quarters of the way through the ordered data

4. Smallest Value

5. Largest Value

The interquartile range is Q3-Q1 and measures the spread of the middle 50% of the data

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A histogram can be used to display grouped continuous data

The area of each bar should be in proportion to the frequency

When the class widths are not all equal, proportional areas can be achieved by plotting the frequency density on the vertical axis

The class width of an interval is calculated as the difference between the smallest and largest values that could occur in that interval

Frequency Density = frequency/class width
For all histograms remember:

Frequency density on the vertical axis

Choose suitable scales for axes

Label both axes

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Population

The total number of things which are available

Sample

Taking a group from the entire population to analyse

Simple Random Sampling

A procedure where every possible element within  a population has an equal chance of being selected, all possible sample groups are equally likely

One way to create a random sample it to allocate all elements within the population a number and use a random number generator such as a calculator to produce a random list of values. Select from the population the elements whose value matches those within the random list.

Systematic Sampling

Taking participants from the population at regular intervals, from a list of the whole population

To create a systematic sample: determine the number of ‘things’ within the population and the sample size which is required and divide the population by the sample size. Then choose a random starting point and, using the previous value, choose every nth element until the sample is built.

Opportunity Sampling

Samples only from those who are available and willing to take part

Care mist be taken as this can introduce bias, especially when groups are small and come from a condensed area