# What are the summary statistics of the test results? Mark - 34,35,36,37,38,39,40,41,42,43 Freq- 2,3,7,1,7,2,3,4,2,0 Find the mean, median, mode, standard deviation, interquartile range and...

What are the summary statistics of the test results?

Mark - 34,35,36,37,38,39,40,41,42,43

Freq- 2,3,7,1,7,2,3,4,2,0

Find the mean, median, mode, standard deviation, interquartile range and z-score of a student in the top 7%.

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### 1 Answer

Mark - 34,35,36,37,38,39,40,41,42,43 Freq-2,3,7,1,7,2,3,4,2Find Mean, median,mode,standard

n = 2 + 3 + 7 + 1 + 7 + 2 + 3 + 4 + 2 + 0 = 31

Mean = sum(mark x frequency)/sum(frequency)

= (34 x 2 + 35 x 3 + 36 x 7 + 37 x 1 + 38 x 7 + 39 x 2 + 40 x 3 + 41 x 4 + 42 x 2 + 43 x 0)/n

= 37.87

Median is (n+1)/2th value (middle value), ie 16th (ordered) datapoint. The 16th datapoint is 38.

Mode is most common value = 36 *and *38 (bimodal - two modes)

Standard deviation = sqrt(variance)

= sqrt(sum((mark-mean)^2 x frequency)/n)

= sqrt((3.87^2 x 2 + 2.87^2 x 3 + 1.87^2 x 7 + 0.87^2 x 1 + 0.13^2 x 7 + 1.13^2 x 2 + 2.13^2 x 3 + 3.13^2 x 4 + 4.13^2 + 5.13^2 x 0)/n)

= 2.34

Lower quartile is (n+1)/4th value, ie 8th datapoint. The 8th datapoint is 36.

Upper quartile is 3(n+1)/4th value, ie 24th datapoint. The 24th datapoint is 40.

Inter-quartile range = upper quartile - lower quartile = 40 - 36 = 4

Z-score is calculated by (mark-mean)/(standard deviation). To invert this for a given percentile of 93, we need to look up the 93rd percentage point of a standard Normal/Gaussian distribution `Phi^(-1)(0.93)` where `Phi^(-1)` is the inverse distribution function. Doing this we find

`Phi^(-1)(0.93) = 1.475`

So the required z-score is 1.475.

**Mean = 37.87, Median = 38, Mode = 36 and 38, Standard deviation = 2.34, IQR = 4, required Z-score = 1.475**