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Standard Deviation, SM-Bank 014

Albert teaches Physics and sets his class a mid-term exam.

The results are summarised in the Stem and Leaf plot drawn below.
 

  1. Determine the median test score of the data set.   (1 mark)

  2. Calculate the standard deviation of this data set, giving your answer to two decimal places.   (1 mark)

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i.    \(\text{Median}\ = 83\)

ii.   \(\text{Std Dev}\ = 12.14\)

Show Worked Solution

i.    \(\text{Dataset values:}\ 62, 65, 78, 79, 83, 89, 94, 96, 97 \)

\(\text{9 data points}\ \ \Rightarrow\ \ \text{Median = 5th value} \)

\(\text{Median}\ = 83\)
 

ii.    \(\text{By calculator (using Statistics mode):} \)

\(\text{Std Dev}\ = 12.139… = 12.14\ \text{(2 d.p.)} \)

Standard Deviation, SM-Bank 008

In a cricket test match, a scorebook recorded the number of runs scored by England's top six batsman.

The scores are summarised in the Stem and Leaf plot below.
 

Determine the standard deviation of the scores, giving your answer correct to one decimal place.   (2 marks)

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\(\text{Std Dev}\ = 6.6\)

Show Worked Solution

\(\text{Runs scored dataset:}\ 8, 12, 15, 21, 23, 27 \)

\(\text{By calculator (using Statistics mode):}\)

\(\text{Std Dev}\ = 6.574… = 6.6\ \text{(1 d.p.)} \)

Standard Deviation, SM-Bank 007

Ms Arnott has seven students in her Ethics class. The results of the most recent exam, completed by the whole class, is summarised in the Stem and Leaf plot below.
 

Determine the standard deviation of the exam results, giving your answer correct to one decimal place.   (2 marks)

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\(\text{Std Dev}\ = 8.8\)

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\(\text{Exam result dataset:}\ 56, 63, 69, 72, 77, 78, 84 \)

\(\text{By calculator (using Statistics mode):}\)

\(\text{Std Dev}\ = 8.84… = 8.8\ \text{(1 d.p.)} \)

Standard Deviation, SM-Bank 006

Seven players in two basketball teams, the Swifties and the Chiefs, recorded how many 3-point baskets they had shot in the last season.

The results are recorded in the two Stem and Leaf plots below.
 

  1. Determine the standard deviation of the Swifties' results, giving your answer correct to one decimal place.   (2 marks)

  2. Without using calculations, explain which data set will have the smallest standard deviation.  (2 marks)

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i.     \(\text{Std Dev}\ = 13.4 \)

ii.    \(\text{The results of the Chiefs have a much smaller range (18 vs 38)}\)

\(\text{and are a much tighter fit around the expected mean.}\)

\(\text{The Chiefs’ results will therefore have a smaller standard deviation.}\) 

Show Worked Solution

i.    \(\text{Swifties’ dataset:}\ 1, 2, 17, 22, 26, 33, 39 \)

\(\text{By calculator (using Statistics mode):}\)

\(\text{Std Dev}\ = 13.43… = 13.4\ \text{(1 d.p.)} \)
 

ii.    \(\text{The results of the Chiefs have a much smaller range (18 vs 38)}\)

\(\text{and are a much tighter fit around the expected mean.}\)

\(\text{The Chiefs’ results will therefore have a smaller standard deviation.}\)