Download Drawing a Box and Whisker Plot

Data Management
In statistics, mode, median and mean are typical values to represent a pool of numerical
observations. They are calculated from the pool of observations.
Mode is the most common value among the given observations. For example, a person
who sells ice creams might want to know which flavour is the most popular.
Median is the middle value, dividing the number of data into 2 halves. In other words,
50% of the observations is below the median and 50% of the observations is above the
median.
Mean is the average of all the values. For example, a teacher may want to know the
average marks of a test in his class.
Mode
The mode of a set of observations is the value that occurs most frequently in the set. A
set of observations may have no mode, one mode or more than one mode.
Ex.:
Find the mode of the following set of scores.
14 11 15 9 11 15 11 7 13 12
Solution: The mode is 11 because 11 occurred more times than the other numbers
If the observations are given in the form of a frequency table, the mode is the value that
has the highest frequency.
Ex.
Find the mode of the following set of marks.
Marks
Frequency
1
6
2
7
3
7
4
5
5
3
Solution: The marks 2 and 3 have the highest frequency. So, the modes are 2 and 3.
Ex.
Find the mode for each of the following frequency table:
The frequency table below shows the weights of different bags of rice.
Weight (kg)
Bags of rice
(Frequency)
Solution:
45
50
55
60
65
70
75
80
8
11
7
10
9
10
12
8
Mode: 75 kg (highest frequency of 12)
Median
Given a set of observations, the median is the middle value among the observations. To
find the median, you must first arrange the observations in an ascending (or descending)
order.
There are 2 possible cases to consider in finding the median.
Case 1 When the number of observations is odd, the median is the middle value.
Ex.
Find the median of the following set of points in a game:
15, 14, 10, 8, 12, 8, 16
Solution: First arrange the point values in an ascending order (or descending order).
8, 8, 10, 12, 14, 15, 16
The number of point values is 7, an odd number. Hence, the median is the value in the
middle position.
Median = 12
Case 2. When the number of observations is even, the median is the average of the two
middle values.
Ex.
Find the median of the following set of points:
15, 14, 10, 8, 12, 8, 16, 13
Solution: First arrange the point values in an ascending order (or descending order).
8, 8, 10, 12, 13, 14, 15, 16
The number of point values is 8, an even number. Hence the median is the average of
the 2 middle values.
Mean
The mean of a set of observations is the average. It is obtained by dividing the sum of
data by the number of observations.
The formula is:
Ex.
Find the mean of the following set of integers.
8, 11, –6, 22, –3
Solution:
Ex.
The set of scores 12, 5, 7, -8, x, 10 has a mean of 5. Find the value of x.
Solution:
Range
The range of a set of numbers is the difference between the least number and the
greatest number in the set.
Ex.
Find the range:
150, 250, 825, 400, 18, 500
Solution:
a) The largest value is 825. The smallest value is 18
Range = largest value − smallest value = 825 − 18 = 807
Worksheet 1
Mean, Median, Mode, Range
Determine the mean, median mode and range for the sets of data below:
______________________________________________________________________
45.6, 54.6, 44.6, 46.5, 66.4, 54.6
Mean = ________ Median = ________ Mode = ________ Range = ________
______________________________________________________________________
1.6, 2.2, 1.6, 3.5, 2.4, 2.2, 3.5, 2.4, 2.2
Mean = ________ Median = ________ Mode = ________ Range = ________
______________________________________________________________________
120, 320, 330, 220, 202, 210, 230, 320, 210, 201, 310, 330, 240, 210, 330, 230
Mean = ________ Median = ________Mode = ________ Range = ________
______________________________________________________________________
54.2, 34.5, 53.4, 45.8, 24.6, 38.6, 26.8, 43.5, 45.2, 42.6
Mean = ________ Median = ________Mode = ________ Range = ________
Drawing a Stem-and-Leaf Plot
Ex.:
Construct a stem-and-leaf plot for the following set of data.
28 13 26 12 20 14 21 16 17 22
17 25 13 30 13 22 15 21 18 18
16 21 18 31 15 19
Solution:
Step 1: Find the least number and the greatest number in the data set.
The smallest number is 12 (1 in the tens place)
The greatest number is 31 (3 in the tens place)
Step 2: Draw a vertical line and write the digits in the tens places from 1 to 3 on the left
of the line. The tens digit form the stems.
Step 3: Write the units digit to the right of the line. The units digits form the leaves.
Step 4: Rewrite the units digits in each row from the least to the greatest.
Step 5: Include an explanation.
Interpreting a Stem-and-Leaf Plot
Ex.
The following stem-and-leaf plot shows the ages of a group of people in a room.
a) How many people were there in the room?
b) Two people have the same age. What is that age?
c) What is the mode, median and mean of the ages?
Solution:
a) We count the number of digits on the right of the line (leaves). There were 12 people
in the room.
b) The two people were 22 years old.
c) The mode is 22.
The median can be obtained from the average 6th and 7th data.
The average of 22 and 24 is 23.
The median age is 23
To get the mean, we have to first get the total ages.
17 + 18 + 19 + 20 + 22 + 22 + 24 + 25 + 26 + 41 + 42 + 44 = 320
The mean = 320 = 26.67
12
Worksheet 2
Stem and Leaf Plots
Draw stem and leaf plot for the given data:
1.
56, 45, 78, 98, 64, 56, 79, 80, 97, 47, 59,
86, 67, 52, 41, 70, 82, 52, 40, 64, 61, 80
2.
101, 127, 134, 123, 107, 111, 118, 129, 148, 142, 115, 104, 123
3
118, 119, 145, 135, 139, 150, 148, 133, 143, 114, 158, 150, 116, 149, 135, 155
4.
83, 86, 92, 13, 28, 53, 49, 28, 49, 66, 50,
81, 18, 27, 68, 81, 45, 53, 21, 17, 55, 80
Drawing a Box and Whisker Plot
Ex. Construct a box plot for the following data:
12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25
Solution:
Step 1: Arrange the data in ascending order (smallest to largest).
Step 2: Find the median, lower quartile and upper quartile
Median is the middle value.
Lower quartile is the middle value of the lower half.
Upper quartile is the middle value of the upper half.
Median = 22
Lower quartile = 12 Upper quartile = 36
***If there is an even number of data items, then we need to get the average of the
middle numbers.***
Step 3: Draw a number line that will include the smallest and the largest data.
Step 4: Draw three vertical lines at the lower quartile (12), median (22) and the upper
quartile (36), just above the number line.
Step 5: Join the lines for the lower quartile and the upper quartile to form a
box.
Step 6: Draw a line from the smallest value (5) to the left side of the box and draw a line
from the right side of the box to the biggest value (53).
Worksheet 3
Quartiles
Determine the lower quartile, median and upper quartile for the sets of data below:
1) 14, 17.5, 13.5, 12, 16, 15.5
2) 63, 175, 239, 50, 12, 252, 192, 120
Lower quartile =
_________
Lower quartile =
_________
Median =
_________
Median =
_________
Upper quartile =
_________
Upper quartile =
_________
________________________________________________________________
3) 92, 88, 65, 82, 92, 102, 82, 39
4) 51.9, 24.1, 32.5, 78.6, 64.5, 39.4
Lower quartile =
_________
Lower quartile =
_________
Median =
_________
Median =
_________
Upper quartile =
_________
Upper quartile =
_________
________________________________________________________________
5) 21.1, 1.12, 11.2, 12.1, 2.11, 2.12, 1.21, 11.2
Lower quartile =_________
Median = _________
Upper quartile = _________
Worksheet 4
1.
Box and Whisker Plots
Draw box and whisker for the given data:
23, 10, 13, 30, 26, 8, 25, 18
Lower Quartile =
Median =
Upper Quartile =
Range =
________________________________________________________________
2.
Draw box and whisker for the given data:
35, 60, 20, 80, 95, 15, 40, 85, 75
Lower Quartile =
Median =
Upper Quartile =
Range =
Population Mean and Sample Mean
The arithmetic mean is the average of a group of numbers and is computed by summing all
numbers and dividing by the number of numbers. The arithmetic mean is also usually just called
the mean.
A population is a collection of persons, objects or items of interest.
A sample is a portion of the whole and, if properly taken, is representative of the whole.
The sample mean is represented by . It is given by the formula
n is
the number of terms in the sample.
Variance
Variance is
a measure of variability, which describes the spread or dispersion of a set of
data.
Sample Variance
The sample variance is denoted by s2. The main use for sample variances is as estimators of
population variances. The computation of the sample variance differs slightly from computation of
the population variance. The sample variance uses n – 1 in the denominator instead of n because
using n in the denominator of a sample variance results in a statistic that tends to underestimate
the population variance. (This is further explained in the video below)
The formula for sample variance is:
Standard Deviation
The standard deviation is a popular measure of variability. The standard deviation is the square
root of the variance.
Sample Standard Deviation
The sample standard deviation is denoted by s.
It is given by the formula
Worksheet 5
Range, Variance, and Standard Deviation
Calculate the values to the nearest tenth.
1.
96, 70, 73, 19, 15, 61, 15, 78, and 84.
Write the range: _________
2.
29, 71, 87, 29, and 14.
Write the range: _________
3.
20.7, -13.2, -2.3, 14.1, 14.1, -9.4, 45.1, -37.7, 27.7, and -12.6.
Write the range: _________
4.
29.1, -38.5, -40.5, 5.1, 18.3, -48, -38.5, 9.2, -6.2, and 12.
Write the range: _________
5.
Given the following prices of used homes (in thousands):
167.1
140.1
179
122.3
102.7
132.1
104.7
117.1
139.5
62.7
140
107.4
118.6
128.4
119.6
184.3
103.1
71.3
108.8
157.2
96.9
142
124.8
105.6
67.4
99.7
164.1
189.2
178.6
144.9
.
Write the
range:
_________
Write the
variance:
Write the
_________ standard
deviation:
_________
6.
Given the following:
104.1, 92.6, 136.3, 170.4, 47, 179.2, 80.9, 72.5, 131.1, 148.4, 186.8, 111, 77, 48.5,
50.6, 108.3, 39.8, 151.3, 156.9, and 159.9.
Write the
range:
7.
_________
Write the
variance:
Write the
_________ standard
deviation:
_________
Given the following annual mutual fund returns:
-49.6
22.93
2.69
-49.61
4.67
14.64
-48.47
11.31
22.37
46
11.61
19.94
2.76
8.42
-43.6
20.84
24.72
9.46
10.85
51.36
-46.33
4.86
22.85
12.59
-43.89
-43.6
15.62
25
5.27
0.5
.
Write the
range:
_________
Write the
variance:
Write the
_________ standard
deviation:
_________