You've got a box-and-whisker plot showing how a class performed on a test, and you're curious about the average score. It's a common question, and thankfully, the information provided gives us a clear path to finding that mean.
We're told that Susan got the highest score, and Tom scored 65. More importantly, we have their standard scores: Susan's is 3, and Tom's is 0.5. Now, standard scores, or z-scores, are fascinating because they tell us how many standard deviations a particular data point is away from the mean. The formula for a standard score is (X - μ) / σ, where X is the individual score, μ is the mean, and σ is the standard deviation.
Let's plug in what we know. For Susan, her score (which we don't know the exact value of, but we know it's the highest) and her standard score of 3 give us an equation. For Tom, we have a concrete score of 65 and a standard score of 0.5. So, we can write two equations:
(Susan's Score - μ) / σ = 3 (65 - μ) / σ = 0.5
Now, the reference material also hints that the highest score in the distribution is 90 marks. This is a crucial piece of information. So, Susan's score is 90.
Our equations become:
(90 - μ) / σ = 3 (65 - μ) / σ = 0.5
We can rearrange these to make them easier to solve. From the first equation, 90 - μ = 3σ. From the second, 65 - μ = 0.5σ.
Now we have a system of two linear equations with two unknowns (μ and σ). A neat way to solve this is to express μ in terms of σ from both equations and set them equal, or to subtract one equation from the other. Let's try subtracting the second from the first:
(90 - μ) - (65 - μ) = 3σ - 0.5σ 90 - μ - 65 + μ = 2.5σ 25 = 2.5σ
Solving for σ, we get σ = 25 / 2.5 = 10.
So, the standard deviation of the distribution is 10 marks.
Now that we have the standard deviation, we can easily find the mean (μ) using either of our rearranged equations. Let's use the second one:
65 - μ = 0.5σ 65 - μ = 0.5 * 10 65 - μ = 5
Rearranging to solve for μ: μ = 65 - 5 μ = 60
And there you have it! The mean of this distribution is 60 marks. It's quite satisfying when all the pieces of information click into place to reveal the underlying structure of the data.
