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

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

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