smc-1131-20-Median and Mode

Statistics, STD1 S1 2023 HSC 13

The graph shows the frequency of scores out of 10 awarded to a museum by visitors.

What is the mode of these data? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Describe TWO features of this graph. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $\text{Mode = Score with the highest frequency}=9$

b. $\text{Features could include any 2 of the following:}$

$\text{Data is negatively skewed as the mean and median are to the left of the mode}$
$\text{23 of the 24 scores, or 95.8%, are 5 or above.}$
$Q_2\text{(Median)}=8$
$Q_1=7\text{ and }Q_3=9$
$IQR=Q_1-Q_3=9-7=2$
$Q_1-1.5\times IQR=7-1.5\times 2=4\ \ \Rightarrow \ \ \text{1 is an outlier}$
$\text{Mean}=\dfrac{189}{24}=7.875$
$\text{If the outlier (1) was removed, Mean}=\dfrac{188}{23}=8.174\text{ (2 d.p.)}$

Show Worked Solution

a. $\text{Mode = Score with the highest frequency}=9$

b. $\text{Features could include any 2 of the following:}$

$\text{Data is negatively skewed as the mean and median are to the left of the mode}$
$\text{23 of the 24 scores, or 95.8%, are 5 or above.}$
$Q_2\text{(Median)}=8$
$Q_1=7\text{ and }Q_3=9$
$IQR=Q_1-Q_3=9-7=2$
$Q_1-1.5\times IQR=7-1.5\times 2=4\ \ \Rightarrow \ \ \text{1 is an outlier}$
$\text{Mean}=\dfrac{189}{24}=7.875$
$\text{If the outlier (1) was removed, Mean}=\dfrac{188}{23}=8.174\text{ (2 d.p.)}$

♦ Mean mark (a) 47%.
♦♦♦ Mean mark (b) 18%.

Statistics, STD2 S1 2018 HSC 11 MC

A set of data is summarised in this frequency distribution table.

Which of the following is true about the data?

Mode = 7, median = 5.5
Mode = 7, median = 6
Mode = 9, median = 5.5
Mode = 9, median = 6

Show Answers Only

`text(B)`

Show Worked Solution

`text{Mode = 7 (highest frequency of 9)}`

`text(Median = average of 15th and 16th data points.)`

`:.\ text(Median = 6)`

`=>\ text(B)`

Statistics, STD2 S1 2004 HSC 6-7 MC

Use the set of scores 1, 3, 3, 3, 4, 5, 7, 7, 12 to answer Questions 6 and 7.

Question 6

What is the range of the set of scores?

Question 7

What are the median and the mode of the set of scores?

Median 3, mode 5
Median 3, mode 3
Median 4, mode 5
Median 4, mode 3

Show Answers Only

`text(Question 6:)\ C`

`text(Question 7:)\ D`

Show Worked Solution

`text(Question 6)`

`text(Range)`	`= text(High) – text(Low)`
	`= 12 – 1`
	`= 11`

`=> C`

`text(Question 7)`

`text(9 scores)`

`:.\ text(Median)`	`= (9 + 1) / 2`
	`=5 text(th score)`
	`= 4`

`text(Mode) = 3`

`=> D`

Statistics, STD2 S1 2007 HSC 24a

Consider the following set of scores:

`3, \ 5, \ 5, \ 6, \ 8, \ 8, \ 9, \ 10, \ 10, \ 50.`

Calculate the mean of the set of scores. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
What is the effect on the mean and on the median of removing the outlier? (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`11.4`
`text{If the outlier (50) is removed, the mean}`

`text(would become lower.)`
`text(Median will NOT change.)`

Show Worked Solution

i. `text(Total of scores)`

`= 3 + 5 + 5 + 6 + 8 + 8 + 9 + 10 + 10 +50`

`= 114`

`:.\ text(Mean) = 114/10 = 11.4`

ii. `text(Mean)`

`text{If the outlier (50) is removed, the mean}`

`text(would become lower.)`

`text(Median)`

`text(The current median (10 data points))`

`= text(5th + 6th)/2 = (8 + 8)/2 = 8`

`text(The new median (9 data points))`

`=\ text(5th value)`

`= 8`

`:.\ text(Median will NOT change.)`

Statistics, STD2 S1 2007 HSC 21 MC

This set of data is arranged in order from smallest to largest.

`5, \ 6, \ 11, \ x, \ 13, \ 18, \ 25`

The range is six less than twice the value of `x`.

Which one of the following is true?

The median is 12 and the interquartile range is 7.
The median is 12 and the interquartile range is 12.
The median is 13 and the interquartile range is 7.
The median is 13 and the interquartile range is 12.

Show Answers Only

`D`

Show Worked Solution

`5, 6, 11, x, 13, 18, 25`

`text(Range)`	`= 2x – 6`
`25 – 5`	`= 2x – 6`
`2x`	`= 26`
`x`	`= 13`
`:.\ text(Median)`	`= 13`

`Q_1 = 6\ \ \ \ \ Q_3 = 18`

`:.\ text(IQR) = 12`

`=> D`

Statistics, STD2 S1 2008 HSC 8 MC

What is the median of the following set of scores?

Show Answers Only

`C`

Show Worked Solution

`text(Median`	`=(n+1)/2`
	`=(33+1)/2`
	`=\ text (17th score)`

`:.\ text(Median is 14)`

`=> C`

Statistics, STD2 S1 2011 HSC 14 MC

A data set of nine scores has a median of 7.

The scores 6, 6, 12 and 17 are added to this data set.

What is the median of the data set now?

Show Answers Only

`B`

Show Worked Solution

`text(S)text(ince an even amount of scores are added below and)`

`text(above the existing median, it will not change.)`

`=>B`

Statistics, STD2 S1 2009 HSC 3 MC

The eye colours of a sample of children were recorded.

When analysing this data, which of the following could be found?

Mean
Median
Mode
Range

Show Answers Only

`C`

Show Worked Solution

`text(Eye colour is categorical data)`

`:.\ text(Only the mode can be found)`

`=> C`

Statistics, STD2 S1 2013 HSC 14 MC

The July sales prices for properties in a suburb were:

$552 000, $595 000, $607 000, $607 000, $682 000, and $685 000.

On 1 August, another property in the same suburb was sold for over one million dollars.

If the property had been sold in July, what effect would it have had on the mean and median sale prices for July?

Both the mean and median would have changed.
Neither the mean nor the median would have changed.
The mean would have changed and the median would have stayed the same.
The mean would have stayed the same and the median would have changed.

Show Answers Only

`C`

Show Worked Solution

`text(Mean increases because new house is sold above)`

`text(the existing average.)`

`text(Initial median)= (607\ 000+607\ 000)/2=607\ 000`

`text(New median)=607\ 000\ \ \ text{(4th value in a list of 7)}`

`=>\ C`