Let a Professor is making a test
Here are the students results (out of 60 points):
20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17
Most students didn't even get 30 out of 60, and most will fail. (marks below 20 conventionally)
The test must have been really hard, so the Prof decides to Standardize all the scores and only fail people 1 standard deviation below the mean.
The Mean is 23 (μ), and the Standard Deviation(σ) is 6.6 by
Standard Deviation": |
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And and standardized score z;
and these are the Standard Scores, which can be treated as relative scores with respect to hardness of the test:
-0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91
Only 2 students will fail (the ones who scored 15 and 14 on the test)
The Standard Deviation is a measure of how spread out numbers are and Normal Distribution is the distribution of data around the mean, which is the hill point of the curve. Distribution of standard deviation around mean is shown in image.
look at the cumulative percentage in orange, it indicates position of a particular data value in entire collection of data, for example if student has got 26 marks in our example, which has Z score 0.45 (let 0.5 ) he stands above 69.1% (50mean+19.1) of students of the class, and 31.9% has higher marks than him.
We have seen an exam could be set very hard in response to increasing competition.One can not fix conventional 33% as passing limit in this case.
This is what exactly happens in case of Graduate Aptitude Test in Engineering, known as GATE, conducted by one of 7 IITs or IISc Banglore every year to provide scale technical and logical abilities of engineering graduates seeking for higher education or job opportunities.
To decide cut-off marks for a particular stream of engineering they choose to go with mean + 1 standard deviation (you can find in GATE information brochure), that means z score +1 is set as cut-off.
As we have seen mean + 1 standard deviation (z score 1)covers 84.1 % (50% for mean and 34 % for +1Standard deviation) of the total distribution , Z score 1 is set as cut-off score, which means cut-off score covers 84% candidates below cut-off marks and select rest 15.9% those are considered to be eligible (passed the exam).
So if cut-off marks for a particular exam is 26 (out of 100), it means only 16 % of candidates appeared in exam got marks more than 26, and rest 84.1% stands below.
An interesting thing to mention here, clueless candidates who are giving exam without preparation will get marks in lower limit of distribution thus pulling down the mean but increases the standard deviation(spread marks), what would be the overall effect on quantity MEAN + STANDARD DEVIATION (i.e. CUT-OFF) due to clueless candidates? Test your math.
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