Sample medians and sample ranges (Edexcel GCSE Statistics): Revision Notes
Sample medians and sample ranges
Introduction to quality control using sample statistics
Quality control charts are powerful tools that help manufacturers monitor the consistency of their products. While we often think about using means (averages) in these charts, we can also use sample medians and sample ranges to track different aspects of production quality. These alternative measures provide valuable insights into both the central tendency and variability of manufactured goods.
The fundamental process remains consistent regardless of which statistic we choose: samples are collected at regular intervals, measurements are taken, and decisions about production adjustments are made based on the results.
Control charts and sample medians
Understanding median-based control charts
When we create a control chart using sample medians, the overall structure looks remarkably similar to charts that use sample means. The key difference lies in what we're plotting on the vertical axis - instead of sample averages, we plot the middle value from each sample.
The plotting process
For each sample collected from the production line, we follow these steps:
- Arrange all measurements in the sample in ascending order
- Find the median value (the middle number)
- Plot this median value on the control chart
- Connect the points to show trends over time
Setting the control limits
The control limits for median charts follow the same sigma rules as mean charts:
- Warning limits: Set at ±2σ from the target median
- Action limits: Set at ±3σ from the target median
These limits help us decide when production might be going off track and when we need to take corrective action.
Control charts and sample ranges
Why monitor sample ranges?
Whilst keeping track of the median or mean tells us about the central tendency of our production, monitoring the sample range gives us crucial information about the variability or spread of our measurements. This is particularly important because it's possible for a production line to maintain the correct average weight (for example) whilst still producing unacceptable variations in individual items.
Understanding variability through ranges
Consider a biscuit factory where the target weight is being met on average, but individual biscuits vary wildly in weight. Some might be far too light whilst others are too heavy. The range helps us spot this problem quickly because it measures the difference between the highest and lowest values in each sample.
Range = Highest value - Lowest value
Advantages of using ranges
The range is particularly useful in quality control because:
- It's much quicker to calculate than the standard deviation
- It gives an immediate sense of how spread out the data is
- Workers on the production floor can calculate it easily without complex mathematics
Special considerations for range charts
When creating control charts for ranges, there's an important practical consideration: sometimes no lower limit is shown. This happens because having zero variation might actually indicate a problem with the testing equipment rather than perfect production quality.
Important relationship with normal distribution
A crucial point to remember is that there's no straightforward relationship between sample ranges and the normal distribution. Unlike means and medians, the warning and action limits for ranges must be determined through the testing process itself rather than using standard sigma rules.
Worked example: Biscuit factory quality control
Let's examine a real-world application of range control charts.
The scenario
A biscuit factory uses a control chart to monitor sample ranges. Looking at their data, we can see several sample points plotted, with most falling within acceptable limits.
Key questions and solutions
Question: Should the production line have been stopped at any point? Answer: No, because all the range values remain within the warning limits throughout the monitoring period.
Calculation example: For sample number 4, the individual weights were: 15.2g, 14.9g, 15.3g, 15.0g, 15.1g, 14.9g, 15.1g
To find the range:
- Identify the highest value: 15.3g
- Identify the lowest value: 14.9g
- Calculate the range: 15.3 - 14.9 = 0.4g
Action needed: Since the range of 0.4g falls within the warning limits, no corrective action is required at this point.
Exam tip
Although calculating ranges is straightforward, always show your working clearly in exams. Write out the highest value, lowest value, and the subtraction explicitly, even when you can do it in your head.
Practice application
Understanding how to set up control limits is essential for applying these concepts in different situations.
Cake production example
Consider a production line making cakes where:
- Target weight: 125g
- Standard deviation (σ): 3.5g
For a median control chart, you would need to calculate:
- Warning limits: 125 ± (2 × 3.5) = 125 ± 7g
- Upper warning limit: 132g
- Lower warning limit: 118g
- Action limits: 125 ± (3 × 3.5) = 125 ± 10.5g
- Upper action limit: 135.5g
- Lower action limit: 114.5g
Remember!
• Sample medians provide an alternative to sample means for monitoring central tendency in quality control, using the same sigma-based limit system
• Sample ranges are essential for monitoring variability and are quicker to calculate than standard deviations
• Warning limits are set at ±2σ from the target for medians, whilst action limits are at ±3σ
• Range calculations are straightforward: simply subtract the lowest value from the highest value in each sample
• No lower limits on range charts sometimes indicate that zero variation might signal equipment problems rather than perfect quality