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### Topic One

Statistical Analysis

Variability of data

- All living things vary, even two peas in the same pod, so how do we measure this variation?
- We plot data usually using the mean, but error bars are a graphical representation of variability of data.

either showing range (highest & lowest value)

or showing standard deviation (±1 s.d.)

What is the importance of error bars?

- The error bars show the spread of data around the mean
- If the error bars are large then the data has a large range and if they are small the range is smaller.
- If we use the sd as our error bars it is more accurate as it includes all data and not just the range, which may be extreme and not representative of the results

1.1.2 Calculate mean and standard deviation of a set of values

- Appreciate the nature of the mean (average) value
- Consider whether mode (most frequent value) or median values (middle value) would be more useful.
- 12 14 11 15 17 10 13 14 14 16
- 12 12 12 12 19 122 147 209 12 22
- 10 11 12 13 14 15 16 17 18 19 20

Mean is useful, it is the focus of the data and mode

We can then measure the deviation or variance from the mean

Evaluation of the variance of the mean useful

What is important about normally distributed data?If the data is tightly clustered there will be small variances, but if the data is more evenly spread over the whole range, the variance would be bigger

Standard Deviation variances, but if the data is more evenly spread over the whole range, the variance would be bigger

- The standard deviation is an 'average' number for the distance of the majority of measures from the mean.
- The standard deviation is usually a preferable method of measuring spread, as opposed to the simpler 'Range' calculation, as it takes account of all measurements.
- The Greek letter, sigma, (s) is often used to signify standard deviation.It is of particular value when used with the Normal distribution, where known proportions of the measurements fall within one, two and three standard deviations of the mean.

This shows the standard deviations about a normal distribution

1.1.3 State-standard deviation summarizes the spread of values around the means and 68% of the values fall within one standard deviation and 95% of values fall within two standard deviations

- Mean =66.6
- Standard deviation= 6.6
- Then you can state that if your data is normally distributed;
- 68% of values will fall between 60.0 and 73.2
- 95% of values will fall between 53.4 and 79.8

REMEMBER THIS…… values around the means and

- STANDARD DEVIATION summarises the SPREAD of data around the mean
- 68% of all values fall within 1 sd of the mean ( 34% above and 34% below)

Visit values around the means and www.mymaths.co.uk

username: west

password: data

Click on GCSE statistics - then standard deviation

Calculate mean and standard deviation from frequency table

How can we show the S.D. in our results? values around the means and

- If we plot the mean, we can then use 1 S.D. on an error bar to show the amount of spread of data.
- This will indicate how accurate the results are.
- Small error bars = accurate results

Assume these bars were for values around the means and ±1 s.d.

Comment on significance

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