Quality Assurance (Edexcel GCSE Statistics)

Distribution of Sample Means

• It is possible to take several different samples from a population, and calculate the mean for each sample

• The set of sample means is more closely distributed than the individual data values from the population

  • e.g. the range of the set of sample means is smaller than the range for the entire population

  • This is because the mean for each sample is in between the biggest and smallest values in the sample

    • and therefore in between the biggest and smallest values in the population

Control Charts

What is quality assurance?

• Quality assurance is a process used to make sure that manufactured products meet required standards

  • e.g. that products sold by weight or volume contain approximately the correct amount of product

    • It is usually impossible for each manufactured item to be exactly correct

    • But they shouldn't vary too much from the target values

• To assure quality, samples are taken at regular intervals

  • and the means, medians and/or ranges of the samples are compared against target values

What is a control chart?

• A control chart is a type of time series chart

  • It allows the calculated results for samples to be recorded

  • And it indicates whether any actions need to be taken

• The horizontal axis will show the sample number (1, 2, 3, 4, 5, ...)

  • The vertical axis will show the sample mean (or sample median or range)

• A horizontal line is drawn corresponding to the target value

  • This is the value (weight, length, etc.) that a 'perfect' item would have

• Horizontal upper and lower action lines are drawn corresponding to the upper and lower action limits

  • If a plotted value is above the upper action line (or below the lower action line)

    • then the manufacturing process should be stopped

    • and any machinery, etc., should be reset to bring it back within the target limits

• Horizontal upper and lower warning lines are drawn corresponding to the upper and lower warning limits

  • If a plotted value is between the two warning lines

    • then everything is assumed to be okay

  • If a plotted value is between a warning line and its corresponding action line

    • then everything might be okay

    • but another sample should be taken right away to make sure there is not a problem

• For a properly functioning manufacturing process,

  • almost all sample values will fall within the action limits

    • So any values outside those limits indicate a likely problem

  • most sample values (about 95%) will fall within the warning limits

    • Any values outside those limits could just be random variation

    • but it is worth checking another sample to make sure

• On a control chart for sample range usually only the upper warning and action limits will be shown

  • This is because the target range is usually zero

    • i.e. every item being the exact target weight or volume, with no variation

  • So it is only a problem if the range becomes too large

• A control chart for sample range can spot problems that a chart using sample mean or median would not

  • e.g. a very large and a very small data value can 'cancel each other out' when calculating the mean

    • But you don't want manufactured items being too different from the target value

How do I find the action and warning limits to use on a control chart?

• You need to be able to calculate the warning and action limits for a control chart using the sample mean

  • If the sample median or sample range is used then the question will tell you the limits to use

• The target value will be the mean () of the population

• The warning and action limits depend on the standard deviation () of the population

  • The warning limits are set two standard deviations above and below the mean

    • 

  • The action limits are set three standard deviations above and below the mean

    • 

• Sample means are normally distributed

  • This means that approximately 95% of sample means will fall within the warning limits

    • So even for a properly functioning process about 1 in 20 sample means will fall outside the warning limits

  • And close to 100% of sample means will fall within the action limits

    • For a properly functioning process hardly any sample means should fall outside the action limits