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MEASURES OF LOCATION

1.	From Tony Modra’s goal kicking figures: 7, 5, 0, 7, 8, 5, 5, 4, 5, and 1 goals in 10 matches, find the mode. The mode is 5, because this value occurred the most often (4 times). This can be interpreted to mean that if one match was selected at random, a good guess would be that Tony would kick 5 goals.
2.	In 12 matches a netball player scored 14, 14, 15, 16, 14, 16, 16, 18, 14, 16, 16, and 14 goals. What is the mode? In this case there are two modes, 14 and 16, because both of them occur the most often (5 times).
3.	In the following data set represents the number of home runs scored by softball player in 14 matches. Find and compare the mean, median and mode.
	In order: 0, 0, 1, 0, 0, 2, 3, 1, 0, 1, 2, 3, 1, 0 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3

NOTE:
For numeric variables the mode is not often used as a measure of central tendency. However, for nominal variables the mode is useful as the mean and median do not make sense


EXERCISES

1.		For the following sets of data find the:
	i)	Mean
	ii)	Median
	iii)	Mode
		a) 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 (to 1 decimal place)
		b) 2, 1, 2, 3, 1, 3, 0, 2, 4, 2, 2
		c) 2.4, 3.9, 1.8, 1.7, 4.0, 2.1, 3.9, 1.5, 3.9, 2.6
		d) 153.8, 154.7, 156.9, 154.3, 152.3, 156.1, 152.3
2.		For the following sets of data find the:
	i)	Mean
	i)	Median
	iii)	Mode
	iv)	Describe briefly the relative positions of the mean, median and mode of each data set.

	a)	x	f		b)	x	f		c)	x	f
		-2	3			6.3	2			1	15
		-1	7			6.4	1			2	5
		0	8			6.5	6			3	3
		1	5			6.6	5			4	1
		2	4			6.7	13			5	2
						6.8	4

3.		For each of the following stem and leaf tables find the:
	i)	Mean
	ii)	Modal-class interval

	a)	Stem	Leaf
		2	2 3 8
		3	1 1 4 2
		4	2 2 3 5 8 9 9
		5	2 4 7 7 8
		6	0 3 2
		7	4
		4|2 represents 42

	b)	Stem	Leaf
		0	2
		0	5 6 8
		1	0
		1	5 5 6 6 7 8 8 9
		2	0 0 0 1 1 2 3 3 3 4 4 4
		2	6 6 7 8 8 9 9
		3	0 4
		3	5 6 7 7 8
		2|2 represents 22

	Year	Increase
	1986	53,377
	1987	52,170
	1988	67,000
	1989	90,332
	1990	72,681
	1991	65,226
	1992	76,777
	1993	83,657
	1994	77,753
	1995	82,892

	a)	Calculate the mean population increase for the years 1986-95.
	b)	Calculate the median population increase for the years 1986-95.
	c)	Do you think the difference in these two measures is significant? Give reasons for your answer, and explain which result gives a better indication of the data’s centre.
	d)	For what purposes would the Queensland Government use measures such as these?
5.		The marks out of 10 for forty students who attempted a maths test were recorded as follows:
		9, 10, 7, 8, 9, 6, 5, 9, 4, 7, 1, 7, 2, 7, 8, 5, 4, 3, 10, 7, 3, 7, 8, 6, 9, 7, 4, 2, 3, 9, 4, 3, 7, 5, 5, 2, 7, 9, 7, 1
	a)	Prepare a frequency table of the scores.
	b)	Using the table, calculate the mean, median and mode.
	c)	How would you interpret these results?
6.		The number of people unemployed at the time of the 1996 Census in Tasmania is given in the table below

	Age group	Number unemployed
	15-19	3,688
	20-24	4,031
	25-34	5,432
	35-44	4,360
	45-54	3,162
	55-64	1,702

	a)	Copy the table and by first finding the mid-point of each interval, calculate the average age of an unemployed person in Tasmania.
	b)	What is the modal-class interval?
	c)	In what age group does the median lie?
	d)	Briefly discuss the comparison between these three results.
	e)	Why do you think the number of unemployed decreases after the age group 25-34?
	f)	How might social welfare organisations use these figures?
7.		A random analysis of 100 married men gave the following distribution of hours spent per week doing unpaid household work.

	Hours	Number of men
	0 - <5	1
	5 - <10	18
	0 - <15	24
	15 - <20	25
	20 - <25	18
	25 - <30	12
	30 - <35	1
	35 - <40	1

		a) Copy the table and include columns to find the end-point of each interval, calculate cumulative frequency and cumulative percentages.
		b) Draw the ogive with cumulative frequency as the y-axis.
		c) From the curve, find an approximate median value. What does this value indicate?
		d) What is the modal-class interval?
		e) Calculate the mean. What does this value indicate?
		f) Briefly describe the comparison between mean, median and mode values.
		g) How might you find out whether women spent more hours per week than men doing unpaid household work?
8.		The 1996 Census table below shows annual income of people aged 15 years or more in Western Australia.

	Income ($)	Persons
	0 - 2,079	114,195
	2,080 - 4,159	44,817
	4,160 - 6,239	45,862
	6,240 - 8,319	139,611
	8,320 - 10,399	114,192
	10,400 - 15,599	148,276
	15,600 - 20,799	123,638
	20,800 - 25,999	121,623
	26,000 - 31,199	103,402
	31,200 - 36,399	73,463
	36,400 - 41,599	59,126
	41,600 - 51,999	68,747
	52,000 - 77,999	56,710

		a) What is the modal-class interval?
		b) Copy the table into your books and include columns to find the upper end-point of each interval, calculate cumulative frequencies and cumulative percentages.
		c) Draw the ogive.
		d) From the curve, give an approximate value for the median annual individual income.
		e) Calculate the mean annual income. (Hint: in the above table, the interval 2,080 - 4,159 actually represents 2,080 - <4,160, so the mid-point is 3,120.)
		f) Describe the comparison of mean, median and mode values.
		g) Which measure gives the most accurate picture of the data’s centre?
		h) What type of organisation would use information such as this?

1.	Measure the height of each student in your class to the nearest centimetre. Are there any outliers? Use an appropriate method to find the mean, median and mode. Compare all three measures. Which value gives the best measure of central location and why? Which organisations or companies would find such statistics useful?
2.	Find out what your school’s student population or your year level’s population has been for the last 10 years. Are there any outliers? Use an appropriate method to find the mean, median and mode. Compare all three measures. Which value gives the best measure of central location and why? How would your school or the Education Department use such statistics?
3.	Find from your school’s records the final scores of your favourite school sport. Collect the scores, both for and against, for the last ten years. (If the data is not available, use data for your favourite sporting team.)
	What was the mean final score, both for and against, for the last ten years?
	What was the median final score, both for and against, for the last ten years?
	Are any of the mean final scores similar to the corresponding median final score?
	What can be said about the distributions given these values?
	What are some of the problems you might come across in trying to use statistics to compare school or other sports teams of the past with those of today?