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Standard Deviation, SM-Bank 011

Mr Ralph has a History class that has completed two separate exams during the term.

The results of each exam are summarised in the histograms below.
 

Without using calculations, explain which test will have the highest standard deviation.  (2 marks)

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\(\text{By inspection, Test B has a higher range than Test A.}\)

\(\text{Test A results look to have a tighter spread of data points}\)

\(\text{about its expected mean, when compared to Test B.}\)

\(\text{Test B will therefore have a higher standard deviation.}\) 

Show Worked Solution

\(\text{By inspection, Test B has a higher range than Test A.}\)

\(\text{Test A results look to have a tighter spread of data points}\)

\(\text{about its expected mean, when compared to Test B.}\)

\(\text{Test B will therefore have a higher standard deviation.}\) 

Standard Deviation, SM-Bank 017

Dataset 1 has mean \(\bar x_1\) and standard deviation \(\sigma_1\).

Dataset 2 has mean \(\bar x_2\) and standard deviation \(\sigma_2\).

Consider the following statement:  If  \(\bar x_1 < \bar x_2\), then  \(\sigma_1 < \sigma_2\).

Is this statement correct? Explain your answer.   (2 marks)

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\(\text{Standard deviation is a measure of how much members}\)

\(\text{of a data group differ from the mean value of the group.}\)

\(\text{It follows that the relative standard deviation between two}\)

\(\text{datasets is not affected if}\ \ \bar x_1 < \bar x_2\).

\(\text{Therefore, the statement is incorrect.}\)

Show Worked Solution

\(\text{Standard deviation is a measure of how much members}\)

\(\text{of a data group differ from the mean value of the group.}\)

\(\text{It follows that the relative standard deviation between two}\)

\(\text{datasets is not affected if}\ \ \bar x_1 < \bar x_2\).

\(\text{Therefore, the statement is incorrect.}\)

Standard Deviation, SM-Bank 015

Rhonda and her friends were surveyed about the number of concerts they have been to in the last 12 months.

Their responses are as follows:

\(3, \ 4, \ 8, \ 10, \ 12, \ 13\)

  1. Determine the median number of concerts from this data set.   (1 mark)

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

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i.    \(\text{Median}\ = 9\)

ii.   \(\text{Std Dev}\ = 3.8\)

Show Worked Solution

i.    \(3, \ 4, \ 8, \ 10, \ 12, \ 13\)

\(\text{6 data points}\)

\(\text{Median}\ = \dfrac{\text{3rd + 4th}}{2} = \dfrac{8+10}{2} = 9 \)
 

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

\(\text{Std Dev}\ = 3.771… = 3.8\ \text{(1 d.p.)} \)

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 012

Seven 40-year old males are asked how many children they have in a survey.

The results are summarised in the histogram drawn below.
 

  1. Determine the mean number of children of the group surveyed, giving your answer to two decimal places.   (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.    \(\text{Mean}\ = 2.57\)

ii.   \(\text{Std Dev}\ = 1.05\)

Show Worked Solution

i.    \(\text{Dataset values:}\ 1, 2, 2, 2, 3, 4, 4 \)

\(\text{Mean}\ = \dfrac{1+2+2+2+3+4+4}{7} = \dfrac{18}{7} = 2.571… = 2.57\ \text{(2 d.p.)} \)
 

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

\(\text{Std Dev}\ = 1.049… = 1.05\ \text{(2 d.p.)} \)

Standard Deviation, SM-Bank 010

A dataset is represented by the dot plot drawn below.
 

Determine the standard deviation of the dataset, giving your answer correct to two decimal places.   (2 marks)

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\(\text{Std Dev}\ = 1.25\)

Show Worked Solution

\(\text{Dataset values:}\ 0, 1, 1, 2, 2, 3, 4 \)

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

\(\text{Std Dev}\ = 1.245… = 1.25\ \text{(2 d.p.)} \)

Standard Deviation, SM-Bank 009

Isa Guha recorded the number of sixes hit in the first nine games of the Women's Big Bash Cricket League over two seasons.

The results are recorded in the two dot plots below.
  

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

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

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i.    \(\text{Std Dev}\ = 1.9\ \text{(1 d.p.)} \)

ii.    \(\text{By inspection, season 1 has a slightly higher range}\)

\(\text{(5 vs 4) and its data points can be seen to be much}\)

\(\text{wider spread about the expected mean value.}\)

\(\text{Season 1 will therefore have a higher standard deviation.}\) 

Show Worked Solution

i.    \(\text{Season 1 dataset:}\ 6, 6, 7, 8, 10, 10, 10, 11, 11\)

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

\(\text{Std Dev}\ = 1.930… = 1.9\ \text{(1 d.p.)} \)
 

ii.    \(\text{By inspection, season 1 has a slightly higher range}\)

\(\text{(5 vs 4) and its data points can be seen to be much}\)

\(\text{wider spread about the expected mean value.}\)

\(\text{Season 1 will therefore have a higher standard deviation.}\) 

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

Show Worked Solution

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

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

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

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

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

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

Standard Deviation, SM-Bank 005

9 students completed two quizzes and the results were summarised in the dot plot diagrams below.
 

  1. Determine the median value of the results of Quiz A.   (1 mark)

  2. Calculate the standard deviation of Quiz A and Quiz B, giving your answers correct to one decimal place.  (2 marks)

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i.     \(\text{Median}\ = 7\)

ii.    \(\text{Std Dev (Quiz A)}\ = 1.3 \)

\(\text{Std Dev (Quiz B)}\ = 0.8\)

Show Worked Solution

i.    \(\text{9 data points}\ \ \Rightarrow \ \ \text{Median = 5th data point}\)

\(\text{Median}\ = 7 \)
 

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

\(\text{Quiz A dataset:}\ 5, 5, 6, 7, 7, 8, 8, 8, 9 \)

\(\text{Std Dev (Quiz A)}\ = 1.33… = 1.3\ \text{(1 d.p.)} \)
 

\(\text{Quiz B dataset:}\ 6, 6, 7, 7, 7, 8, 8, 8, 8 \)

\(\text{Std Dev (Quiz B)}\ = 0.78… = 0.8\ \text{(1 d.p.)} \)

Standard Deviation, SM-Bank 004

Two data sets are represented in the dot plot diagrams below.
 

  1. Calculate the standard deviation of Data Set B, giving your answer correct to one decimal place.   (1 mark)

  2. Without using calculations, explain which data set has the highest standard deviation.  (2 marks)

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i.    \(\text{Std Dev}\ = 1.79… = 1.8\ \text{(1 d.p.)} \)

ii.   \(\text{By inspection, Data Set A has a smaller range and a tighter}\)

\(\text{spread against its expected mean value.}\)

\(\text{Data Set B will therefore have a higher standard deviation.}\) 

Show Worked Solution

i.    \(\text{Data points:}\ 4, 4, 5, 5, 6, 6, 7, 8, 9, 9\)

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

\(\text{Std Dev}\ = 1.79... = 1.8\ \text{(1 d.p.)} \)
 

ii.   \(\text{By inspection, Data Set A has a smaller range and a tighter}\)

\(\text{spread against its expected mean value.}\)

\(\text{Data Set B will therefore have a higher standard deviation.}\) 

Standard Deviation, SM-Bank 003

In a small business, the seven employees earn the following wages per week:

\(\$300, \ \$490, \ \$520, \ \$590, \ \$660, \ \$680, \ \$970\)

  1.  Calculate the standard deviation for this set of data, giving your answer to one decimal place.  (1 mark)

  2.  Each employee receives a $20 pay increase.

     

     Explain the effect that this increase will this have on the standard deviation?  (2 marks)

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i.    \(191.0\)

ii.    \(\text{All values increase by \$20, but so too does the mean.} \)

\(\text{Therefore the spread about the new mean will not change} \)

\(\text{and therefore the standard deviation will remain the same.} \)

Show Worked Solution

i.    \(\text{By calculator (in Statistics mode:}\)

\(\text{Std Dev}\ = 191.044...  =191.0\ \text{(1 d.p.)} \)


ii.    \(\text{All values increase by \$20, but so too does the mean.} \)

\(\text{Therefore the spread about the new mean will not change} \)

\(\text{and therefore the standard deviation will remain the same.} \)

Standard Deviation, SM-Bank 002

Ali’s class sits a Geography test and the results are recorded below.

\(58,\ \ 74,\ \ 65,\ \ 66,\ \ 73,\ \ 71,\ \ 72,\ \ 74,\ \ 62,\ \ 70\)

The mean for the test was 68.5. 

  1. Calculate the standard deviation for the test. Give your answer correct to one decimal place.    (2 marks)

  2. Explain whether the mean of the test is greater than or less than the median of the test.    (1 mark)

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i.    \(5.2\ \ \text{(to 1 d.p.)} \)

ii.    \(\text{Median}\ = \dfrac{70+71}{2} = 70.5 \)

\(\Rightarrow\ \text{Median > Mean  (70.5 > 68.5)} \)

Show Worked Solution

i.    \(\text{By calculator (in Statistics mode):}\)

\(\text{Std Dev}\ = 5.220… = 5.2\ \text{(1 d.p.)} \)

♦ Mean mark (a) 39%.

 
ii.    \(\text{Reorder test results in ascending order:}\)

\(58,\ \ 62,\ \ 65,\ \ 66,\ \ 70, \ \ 71,\ \ 72,\ \ 73,\ \ 74,\ \ 74\)

\(\text{Median}\ = \dfrac{70+71}{2} = 70.5 \)

\(\Rightarrow\ \text{Median > Mean  (70.5 > 68.5)} \)

Standard Deviation, SM-Bank 001

The ages of nine students were recorded in the table below.
 

 

  1. What is the standard deviation, correct to two decimal places?   (2 marks)

  2. Briefly explain what is meant by the term standard deviation.   (1 mark)

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i.    \(\text{By calculator (in Statistics mode):} \)

\(\text{Std Dev}\ = 1.5947… = 1.59\ \ \text{(to 2 d.p.)} \)

 

ii.   \(\text{Standard deviation is a measure of how much members}\)

\(\text{of a data group differ from the mean value of the group.}\)

Show Worked Solution

i.    \(\text{By calculator (in Statistics mode):} \)

\(\text{Std Dev}\ = 1.5947… = 1.59\ \ \text{(to 2 d.p.)} \)

 

ii.   \(\text{Standard deviation is a measure of how much members}\)

\(\text{of a data group differ from the mean value of the group.}\)

Statistics, STD1 S1 2020 HSC 24

  1. The ages in years, of ten people at the local cinema last Saturday afternoon are shown.

\(38 \ \ 25 \ \ 38 \ \ 46 \ \ 55 \ \ 68 \ \ 72 \ \ 55 \ \ 36 \ \ 38\)

  1. The mean of this dataset is 47.1 years.
  2. How many of the ten people were aged between the mean age and the median age?  (2 marks)

  3. On Wednesday, ten people all aged 70 went to this same cinema.
  4. Would the standard deviation of the age dataset from Wednesday be larger than, smaller than or equal to the standard deviation of the age dataset given in part (a)? Briefly explain your answer without performing any calculations.  (2 marks)

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a.     \(1\)

b.    \(\text{Standard deviation is a measure of how much the}\)

\(\text{ages of individuals differ from the mean age of the group.}\)
 

\(\Rightarrow\ \text{Standard deviation of Wednesday’s group would be}\)

\(\text{less as the mean is 70 and everyone’s age is 70.}\)

Show Worked Solution

a.     \(\text{Reorder ages in ascending order:}\)

    \(25, 36, 38, 38, 38, 46, 55 , 55, 68, 72\)

\(\text{Median} = \dfrac{\text{5th + 6th}}{2} = \dfrac{38 + 46}{2} = 42\)

\(\therefore\ \text{People with age between 42 − 47.1 = 1}\)

♦ Mean mark (a) 39%.

 
b.    \(\text{Standard deviation is a measure of how much the}\)

\(\text{ages of individuals differ from the mean age of the group.}\)
 

\(\Rightarrow\ \text{Standard deviation of Wednesday’s group would be}\)

\(\text{less as the mean is 70 and everyone’s age is 70.}\)

♦♦♦ Mean mark (b) 20%.

Statistics, STD2 S3 2017 HSC 29d*

All the students in a class of 30 did a test.

The marks, out of 10, are shown in the dot plot.
 

  1. Find the median test mark.  (1 mark)

  2. The mean test mark is 5.4. The standard deviation of the test marks is 4.22.
  3. Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean.  (2 marks)
Show Answers Only
  1. \(6\)
  2. \(\text{43%}\)
Show Worked Solution

i.    \(\text{30 data points}\)

\(\text{Median}\ = \dfrac{\text{15th + 16th}}{2} = \dfrac{4+8}{2} = 6\)
 

♦ Mean mark 50%.

ii.    \(\text{Lower limit}\ = 5.4-4.22 = 1.18\)

\(\text{Upper limit} = 5.4 + 4.22 = 9.62\)

\(\text{% between}\) \(= \dfrac{13}{30} \times 100\)  
  \(= 43.33… \%\)  
  \(=43\%\ \ \text{(nearest %)}\)  
♦♦ Mean mark 34%.

Statistics, STD2 S1 2017 HSC 27a

Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.

`220, 105, 101, 450, 37, 338, 151, 205`

  1. Calculate the mean of this set of data.  (1 mark)

  2. What is the standard deviation of this set of data, correct to one decimal place?  (1 mark)

Show Answers Only
  1. `200.875`
  2. `127.4\ \ text{(1 d.p.)}`
Show Worked Solution
i.   `text(Mean)` `= (220 + 105 + 101 + 450 + 37 + 338 + 151 + 205) ÷ 8`
    `= 200.875`
♦ Mean mark part (ii) 47%.
IMPORTANT: The population standard deviation is required here.

 

ii.   `text(Std Dev)` `= 127.357…\ \ text{(by calc)}`
    `= 127.4\ \ text{(1 d.p.)}`