Mathematics Grade 11
Lesson Notes
Lesson
1
Teacher Guide
Data Handling
Box and whisker plot
In this lesson we revise the use of measures of central tendency (mean, median and mode) and
measures of spread or dispersion (the range, the interquartile range and the semi-interquartile range)
to describe sets of data. We then work with the five-number summary, and the box and whisker plot
as a representation of the five number-summary.
Lesson Outcomes
!
By the end of this lesson, you should be able to:
• work out measures of central tendency
• work out measures of dispersion
• define the five number summary
• draw a box and whisker plot.
Curriculum Links
LO 4: Data Handling and Probability
11.4.1 (a) Calculate and represent measures of
central tendency and dispersion in univariate
numerical data by the five number summary
(maximum, minimum and quartiles) and box and
whisker diagrams.
Lesson Notes
There are three measures of central tendency, the mean, the median and the mode.
• Calculate the mean by adding all the data values together and dividing by the number of data
values.
• Find the median by arranging the data in order, and then finding the middle value. If there is an
even number of values, the median is the average of the two middle data values.
• The mode is the value that occurs most often in a set of data.
The measures of dispersion give us information about how spread out the data is.
• The range is the difference between the highest value (or maximum) and the lowest value (or
minimum) in a data set. It shows how spread out the whole data set is.
• The quartiles and the median divide the data set into four parts. The lower quartile divides
data below the median into two equal quartiles and the upper quartile divides the data above
the median into two equal quartiles. The difference between the lower and the upper quartile is
called the interquartile range. This tells us how spread out the middle half of the data is around
the median. The semi-interquartile range is half of the interquartile range.
• In the data set below, the range is 80, and the interquartile range is 16.
minimum
2
median
12
lower
quartile
18
28
maximum
82
upper
quartile
Interquartile range
The five number summary, which we can use to describe a data set, consists of the minimum and
maximum values, the median and the quartiles. We represent these numbers on a box and whisker
plot. The box represents the middle half of the data, with the line in the box showing the median. The
‘whiskers’ show the minimum and maximum values. (It is also acceptable to limit each whisker to a
length one and a half times the length of the box.)
1
Mathematics Grade 11
Lesson Notes
Lesson
1
Teacher Guide
Data Handling
box & whisker plot:
2
0
12
5
16
18
28
82
whisker
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
A box and whisker diagram indicates whether a data set is symmetrical or whether it is positively
or negatively skewed. The diagram above is not symmetrical. If the data were evenly spread, the
whiskers would be the same length and the median would be in the middle of the box, giving the
whole diagram a symmetrical appearance. In this case, the whisker on the right is much longer. So
although there are the same number of data values on both whiskers, they are more spread out on
the right. We say that the data is positively skewed. The median is also slightly to the left in the box,
which also indicates positive skewing.
Task
? coffee shop in the hospital wants to know how many bread rolls to order each day so that they
The
box & whisker plot:
2
0
12
5
16
18
28
box & whisker plot:
whisker
2
82
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
0
12
5
16
18
28
62
75
82
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
don’t run out of rolls or have too many unsold at the end of the day.
The data shows the number of bread rolls sold per day in two weeks. Find the mean, the median,
and the upper and lower quartiles. Then calculate the range and the interquartile range. Use these
measures to recommend what number of bread rolls should be ordered daily.
Number of bread rolls sold per day:
72, 65, 56, 56, 48, 47, 36, 37, 48, 49, 55, 55, 64, 73
2