Comparing a value with mean and standard deviation

Question

See the following table,

           Mean    Standard Deviation   Marks of Tom 
English     65            10                  55
Maths       51             4                  59
Science     65             4                  65
History     82             6                  64

By using the given details what can you tell about the level of performance of each subject and level of performance of Tom. Plz Help.

Answer 1

It would help if you gave the context of the problem. My guess is that you are studying 'standardization.'

Given raw score $X$, population mean $\mu$ and population standard deviation (SD) $\sigma.$ The standard score (or z-score) is $Z = (X - \mu)/\sigma.$ If you do not know $\mu$ and $\sigma$ in a particular application, you might estimate $\mu$ by a sample mean and $\sigma$ by a sample SD. From what you say, there is no way to know whether you have sample or population means and SDs. Maybe they are summary statistics for exams Tom recently took in four subjects.

In the first part of your problem, Tom's raw score 55 has standard score $Z_{Eng} = (55 - 65)/10 = -1.$ That is, Tom's English score falls one SD below the mean.

Similarly, $Z_{Math} = (59 - 51)/4 = 2,$ two SDs above the mean.

One might say that Tom did considerably 'better' compared to the 'competition' in Math than in English. [Without giving formulas, @Karl's Answer is hinting at the same idea. (+1)]

I will leave it to you to find Tom's standard scores for Science and History.

Later in the course, you may be dealing with normally distributed data and use standardization as a way to find normal probabilities from printed tables of the standard normal distribution.

Answer 2

To answer this question ask yourself, "which subject did Tom do worse in English or History?". In both of this subjects he is below average. To answer this you should consider the standard deviation or average displacement from the mean as it tells you the spread of data for that subject. Immediately you find that Tom is one standard deviation below the mean in English and over two standard deviations below that in History. Tom is comparatively better at English as two standard deviations below is more of an extreme result.

Note he could be taking harder tests so he is really only compared to his peers in the same subject and not really himself in across the subjects.

For further info look up standard scores.

Hope it helps.