Introduction to Standardised Scores

Have you ever tried to figure out which of your friends is better at sports when one plays football and the other runs track? Or have you ever wondered if your 75% in a "hard" Maths test is actually better than your 85% in an "easy" English test?

In Statistics, we call this the "Apples and Oranges" problem. You can’t compare them directly because they come from different groups with different averages and spreads. Standardised scores (often called z-scores) solve this by putting different sets of data on the same "playing field."

Note: This is a Higher Tier only topic. If you are sitting the Foundation paper, you do not need to learn this formula!

What is a Standardised Score?

A standardised score tells us how many standard deviations a particular value is away from the mean. Instead of looking at the "raw" score (the actual number), we look at how that score relates to the rest of its group.

By standardising data, we can compare values from two completely different samples, even if they have different units or scales.

The Formula You Need to Know

In the Pearson Edexcel GCSE Statistics exam, the formula for a standardised score is not given on the formula sheet. You must memorise it:

Standardised score = \( \frac{x - \text{mean}}{\text{standard deviation}} \)

Where:
\(x\) = the individual value or "raw score" you are looking at.
mean = the average of the data set.
standard deviation = the measure of how spread out the data is.

Quick Tip for the Calculator:

When typing this into your calculator, always calculate the top part (\(x - \text{mean}\)) first, or put it in brackets. If you don't, your calculator might only divide the mean by the standard deviation because of BIDMAS!

Interpreting the Results

Once you calculate the standardised score, the number tells you a specific story about that piece of data:

  • A score of \(0\): The value is exactly equal to the mean (perfectly average).
  • A positive score (e.g., \(+1.5\)): The value is above the average.
  • A negative score (e.g., \(-2.1\)): The value is below the average.
  • The size of the number: The further the score is from \(0\), the more "unusual" or "extreme" it is compared to the rest of the group.

Step-by-Step Example: Who performed better?

Imagine two students, Sam and Alex, want to know who did better relative to their class.

Sam (Maths): Score = \(70\), Class Mean = \(60\), Class SD = \(5\)
Alex (English): Score = \(80\), Class Mean = \(75\), Class SD = \(10\)

Step 1: Calculate Sam's standardised score.
\( \text{Sam's score} = \frac{70 - 60}{5} = \frac{10}{5} = +2.0 \)

Step 2: Calculate Alex's standardised score.
\( \text{Alex's score} = \frac{80 - 75}{10} = \frac{5}{10} = +0.5 \)

Step 3: Compare.
Even though Alex had a higher raw score (\(80\) vs \(70\)), Sam performed better relative to their class because Sam's standardised score (\(+2.0\)) is much higher than Alex's (\(+0.5\)). Sam is \(2\) standard deviations above the mean, while Alex is only half a standard deviation above the mean.

Standardised Scores and the Normal Distribution

Standardised scores are closely linked to the Normal Distribution (the "Bell Curve"). As you may have learned in the probability section, in a normal distribution:

  • About \(68\%\) of data falls between standardised scores of \(-1\) and \(+1\).
  • About \(95\%\) of data falls between standardised scores of \(-2\) and \(+2\).
  • Almost all data (\(99.7\%\)) falls between \(-3\) and \(+3\).

Did you know? If you get a standardised score higher than \(+3\) or lower than \(-3\), it is often considered an outlier because it is so far from the mean!

Common Mistakes to Avoid

1. Swapping the formula: Students often do \( \text{mean} - x \) by mistake. Always start with the individual value (\(x\)). If you are below the mean, your result must be negative.

2. Forgetting the units: Standardised scores do not have units (like cm or kg). They are just pure numbers that show "distance in standard deviations."

3. Misinterpreting "Better": In some contexts, a lower raw score is better (like a golf score or a race time). In those cases, the person with the more negative standardised score actually performed better!

Key Takeaways Summary

Standardised Score = \( \frac{x - \text{mean}}{\text{SD}} \)

  • Use it to compare scores from different data sets.
  • It measures how many standard deviations a value is from the mean.
  • A positive score is above average; a negative score is below average.
  • Standardised scores make data comparable by removing the original units and scales.
Quick Review:

If a student gets a score of \(55\) on a test where the mean is \(55\), what is their standardised score?
Answer: \(0\). Because \( \frac{55 - 55}{\text{SD}} = 0 \). They are exactly on the mean!