Name: _______________________
Class: __________
Date:_____________
Math SL: 14B Measuring the center of
data
14B.1 Mean, median and mode of UNGROUPED data
Today’s Objective:
(1) to review concepts like mean, median and mode and their merits. Consideration for
ungrouped data vs. grouped data needs to be taken into account.
Knowing about the center of a data set provides a better understanding of the data. There are
three different measures of center.
1.
Mode – used for discrete numerical data, it is the most frequently occurring value in the data
set.
Modal class – used for continuous numerical data, it is the class (interval) that occurs most
frequently.
For a discrete data set, there can be one mode, more than one mode or no mode.
A data set with two modes is called bimodal.
If a data set has three or more modes then it is not used a measure of the middle of the data.
2.
Mean – also called the arithmetic average.
mean =
sum of all data values
number of data values
k
=
å xi
i=1
n
x is the mean of a ____________
m is the mean of a __________________
3.
Median – this is the middle value of any data set when the data are ordered from smallest to
largest.
The median splits the data into halves.
For an odd number of data, the median is one of the original data values.
For an even number of data, the median is the average of the two middle values and may
not be in the original data set.
æ n + 1ö
÷ th data value.
2 ø
From Textbook: If there are n data values, the median is the ç
è
1
For distributions that are symmetric, the mean or median will be approximately ___________.
Both measures accurately measure the center of the distribution.
For distributions that are skewed the ____________ is “dragged” towards the “tail”.
Therefore the ________________ is the best measure of the center of the distribution.
__________________ skewed
__________________ skewed
Summary of Measures of Central Tendency
Mode
Mean
Median
Advantages
 Not affected by extreme values (i.e.
it is a resistant statistic)








Most popular measure
Uses all data values
There is only one mean
Useful when comparing sets of data
Not affected by extreme values
There is only one median
Useful when comparing sets of data
50% of the data is either side of the
median
2
Disadvantages
 Does not use all data values
 May be more than one mode
 Difficult to interpret when there is
more than one mode
 May not exist
 Affected by extreme values (i.e. it is
a non-resistant statistic)

Not used much in further
calculations
Examples:
1. Consider the following two sets of data:
Data set A: 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 10 Data set B: 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 15
a.
Find the mean and median for both data sets.
b.
Explain why the mean of data set A is less than the mean of data set B.
c.
Explain why the median of data set A is the same as the median of data set B.
2.
The mean monthly sales for a clothing store total $15 467. Calculate the total sales for the
store for the year.
3.
After 8 matches, a basketball player had a mean score of 27 points. After 3 more matches her
average was 29. How many points did she score in the last 3 games?
14B.2 Mean, median and mode of GROUPED data
If the same data appear several times we can summarize the data in a table.
Ungrouped Data
Mode – look at the frequency column to determine which data value occurs the most often.
In the example the mode is _______.
Median – sum the frequency column to find n, the total number of values. The median is the
th
æ n + 1ö
çè
÷ value. Add the frequency until you get to the median value.
2 ø
In the example the median is the ________th value, which is _______.
Mean – add a product (frequency × data value) column. The sum of this column is
the total of all the data values. The formula for the mean is now:
3
k
x=
åfx
i i
i=1
k
åf
i
i=1
=
where k is the number of different data values
å fx
åf
In the example the mean =
Example 1: For the data displayed in the stem-and-leaf plot find the mean, median, and mode.
Grouped Data
When information has been gathered in classes we don’t know the individual data values.
We can state the modal class and the group which contains the median value.
To estimate the mean, we use the ____________________ of the class to represent all scores within
that interval. By doing this, we are assuming that the data values within each class are evenly
distributed throughout the interval. The mean calculated is an approximate value and we
cannot do better than this without knowing the individual data value.
Example 2: 50 students sit a mathematics test. Given the results in the table below, estimate the
mean score.
Hmwk35
14B.1 Measuring the center of data (ungrouped data)
pg.386 # 1(b), 4, 5, 7, 9, 12, 13, 14
14B.2 Measuring the center of data (frequency table)
pg.390 # 4, 5(b), 7(b), 9, 10
pg.393 # 2, 3
4