Download Section 3.3 - Salesianum School

STATISTICS 3.3 MEASURES OF CENTRAL TENDENCY
MEAN: or the AVERAGE of a set of numbers is obtained by adding the numbers together and dividing the sum
by number of numbers added.
Symbol: x
 x  x1  x2  x3  ...  xn
SAMPLE MEAN  x 
n
n
MODE: is the number that occurs the MOST often
MEDIAN: is the MIDDLE NUMBER when the numbers are arranged in order from smallest to largest.
EXAMPLE 1: Ms. Orga surveyed three students, Pat, Tom, and Buddy. The following chart indicates the
number of minutes spent watching television daily by each student on seven randomly selected days.
Pat Kirby
Tom Riley Buddy Reynolds
120
175
30
75
175
45
10
90
85
75
70
105
110
70
150
140
50
60
120
125
135
Calculate the sample mean, median, and mode for each student.
2) Bob Sabol has a bit of a heavy foot. The following shows how fast he was driving to school. Make a
box plot on your calculator and copy it onto the sheet.
65
52
67
45
68
59
70
61
56
67
74
68
66
70
73
71
76
74
Ordered data: (CIRCLE the median)
Lower half: (CIRCLE the Lower Quartile – the median of the lower half)
Upper half: (CIRCLE the Upper Quartile – the median of the upper half)
78
84
STATISTICS 3.3 MEASURES OF CENTRAL TENDENCY
EXAMPLE 2: Calculate the sample mean, median, and mode of the following numbers:
28
19
25
17
29
19
26
17
28
15
17
28
31
22
31
17
31
28
31
14
31
14
17
28
24
31
17
14
26
24
24
24
19
24
14
28
22
31
17
22
25
12
26
19
26
12
19
26
19
28
The sample mean of a distribution of grouped data is given by:
x
 xf
f
Number
x
15
Frequency
f
1
17
7
19
6
22
3
24
5
25
2
26
5
28
7
29
1
Product
xf
=
=
The sample mean for this frequency distribution is:
Number
Class Mark
x
15-17
Frequency
f
8
18-20
6
21-23
3
24-26
12
27-29
8
30-32
7
=
The sample mean for this frequency distribution is:
Product
xf
=
STATISTICS 3.3 MEASURES OF CENTRAL TENDENCY
If w1 , w2 ,..., wn are the weights assigned to the numbers x1 , x2 ,..., xn then the weighted sample mean, (xw ), is
xw 
 xw  x w  x w  ...  x w
w  w  ...  w
w
1
1
2
1
2
2
n
n
n
EXAMPLE 3: The grades Mike got in his statistics class and the weight assigned to each are as follows.
Test 1
Test 2
Test 3
Test 4
Final
Find Mike’s average term grade.
Grade
84
73
62
91
96
Weight Assigned
1
2
5
4
3
EXAMPLE 4:
Test Failures. Dan LaRosa is one of Ms. Orga’s most consistent students. It is getting towards the end of
the year and he got the following scores on his tests.
Year
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
Test 8
Test 9
Test 10
Grade
106
78
85
79
81
42
83
80
79
82
This data contains two extremes (or OUTLIERS). For situation like this a statistician will compute a
trimmed mean where high and low values are excluded or “Trimmed off” before calculating the mean. In
this case we are calculating the trimmed mean by excluding the top 10% and bottom 10%.
What is his trimmed mean?
STATISTICS 3.3 MEASURES OF CENTRAL TENDENCY
RANGE: is found by finding the difference between the highest and lowest value
Range  Highest  Lowest
MIDRANGE: is found by taking the average of the lowest value L and the highest value H. Thus
LH
Midrange 
2
1. Calculate the range and midrange for the sample
3, 7, 11, 15, 16, 18, 21, 22, 23
2. Calculate the range and midrange for the sample
104, 33, 66, 79, 42, 55, 112
3. The weights of 9 students, measured in pounds, are recorded below.
155,
a.
180,
176,
199,
161,
165,
185,
Find the mean weight.
b. Find the range of weights?
c. Find the mode of the weights?
d. Find the median of the weights?
e. Find the midrange of the weights?
f. Construct a box and whisker plot to describe the data.
171,
223