Video Transcript
In this video, we will learn how to
find three measures of central tendency, or average. They are the mean, the median, and
the mode. We will begin by defining all three
terms. We will then look at some questions
where we need to calculate the mean, median, and mode from a data set.
We will begin by defining the
mean. To calculate the mean of a
numerical data set, we add up all the data values and then divide the total by the
number of values in the data set. For example, if we had the set of
numbers four, three, seven, six, and five, we would firstly add all the values to
give us a total of 25. As there were five values, the mean
would be 25 divided by five, which is equal to five.
Let’s now consider the median. To work out the median of a data
set, we follow two steps. Firstly, we organize the data
according to size in either ascending or descending order. Secondly, we count the values. If the number of values is an odd
number, then the median is the middle value. If the number of values is even,
then the median is the mean of the two values at the center of the ordered data
set.
Consider the set of numbers four,
nine, eight, six, and three. Putting these numbers in ascending
order from smallest to largest gives us three, four, six, eight, and nine. As there are five numbers in total,
the third number in ascending order is the median, in this case, six. If our data set had had an extra
number 10, then our list in ascending order would be three, four, six, eight, nine,
10. This time, our data set has an even
number of values, so there are two middle numbers, six and eight. The median can be calculated by
finding the midpoint of these two values, or the mean of these two values. The answer, in this case, is
seven.
Finally, let’s consider the
definition of the mode. The mode is the most commonly or
frequently occurring value or values. We sometimes call this the modal
value. If we consider the data set four,
seven, six, seven, eight, and two, we notice that the number seven appears
twice. Therefore, this is the mode or
modal value. A data set may have one mode, more
than one mode, or no mode at all. If the data set has two modes, we
say it is bimodal.
A data set can only have one mean
and one median. Whilst there are many different
kinds of mean, the one that we are using here, and the one most commonly used, is
called the arithmetic mean. We will now look at a question
where we will determine the mean, median, and mode for a data set.
The scores earned on a math test
are 86, 80, 76, 68, 73, 85, 74, 70, 71, and 70. Find the mean, median, and mode for
the set of data.
We recall that in order to
calculate the mean from a data set, we firstly need to add all the values. In this question, this is the sum
of the scores in the test. We then need to divide this total
by the number of values, in this case, 10. The sum of the scores is 753. 753 divided by 10 is 75.3. This means that the mean of the
scores is 75.3 marks.
We can calculate the median by
finding the middle value. Before doing this, we must order
the numbers from smallest to largest or largest to smallest. In ascending order, the scores are
68, 70, 70, 71, 73, 74, 76, 80, 85, and 86. As they are an even number of
values, in this case, 10, there will be two middle numbers. These are 73 and 74. The median will be the value
halfway between these two numbers, or the mean of the two numbers. This is equal to 73.5. Therefore, the median score is
73.5.
The mode is the most frequently
occurring value or values in our data set. In this question, eight of the
values appear once, and 70 appears twice. As 70 occurs more often than any
other data value, the mode of the scores is 70. This set of data has a mean of
75.3, a median of 73.5, and a mode of 70.
We will now look at a question
where we can find the mean, median, and mode from a bar graph.
The bar chart shows the yearly
membership of a robotics club from 2001 to 2005. Find the mean, median, mode, and
range of the data.
We can see from the bar chart that
we have five values, 49, 31, 31, 29, and 50. These are the number of members in
each year from 2001 to 2005. The first part of this question
asks us to calculate the mean. To find the mean of a set of
values, we firstly find the sum of the values. In this case, we add 49, 31, 31,
29, and 50. We then divide this total by the
number of values, which in this case is five. The sum of the values is 190. We need to divide this by five. This is equal to 38. So, the mean number of members is
38.
The second part of the question
asks us to find the median value. In order to do this, we firstly
list the values in ascending or descending order. The median is the middle value. And as there are five values here,
the median will be the third value. The median number of members in the
club is 31.
Next, we need to work out the
mode. This is the most common or most
frequently occurring number. The numbers 49, 29, and 50 appear
once, whereas the number 31 appears twice. This means that 31 is the mode of
the data set.
Finally, we’re asked to calculate
the range. This is the highest value minus the
lowest value. As the highest value is 50 and
lowest value 29, we need to subtract 29 from 50. This is equal to 21. The mean, median, mode, and range
of members in the robotics club are 38, 31, 31, and 21, respectively.
We will now look at a question
where we need to select a data set with a given mode and median.
Which of the following sets of data
has a mode of 48 and a median of 20. Is it A) 48, 21, 11, 48, 20,
17? B) 21, 48, 19, 48, 17, 11? C) 47, 47, 11, 48, 20, 17? D) 10, 16, 19, 21, 47, 47? Or E) 20, 48, 48, 11, 11, 19?
We recall that the mode is the most
frequently occurring value. Therefore, we need to find a set of
data where 48 is the most common or most frequently occurring number. 48 does not occur in set D. Therefore, this cannot be the
correct answer. Whilst there is a 48 in set C,
there are two 47s. Therefore, the mode of set C is
47. We can, therefore, rule this out as
the correct answer.
Set E has two 48s, but it also has
two 11s. This means that it has two modes,
or it’s bimodal. It has a mode of 11 and 48. This means that option E is also
incorrect. Both option A and option B have two
48s. This is the most frequently
occurring value in both data sets. This means that the mode of both of
these is 48.
Let’s now consider our second piece
of information. The median of the data set has to
be 20. We know that the median is the
middle value once the numbers are in ascending or descending order. Once we have put both of these data
sets in order, we notice that they have an even number of values, in this case,
six. This means that there are two
middle members, in set A, 20 and 21 and in set B, 19 and 21.
The median can be calculated by
finding the mean of these two values. This is the same as finding the
midpoint of the two values. The mean of 20 and 21 is 20.5. And the mean of 19 and 21 is
20. This means that set A has a median
of 20.5 and set B has a median of 20. We can, therefore, rule out set
A. The set of data that has a mode of
48 and a median of 20 is set B, 21, 48, 19, 48, 17, and 11.
Our next question will look at what
happens to the mean, median, and mode when removing a data value.
Last month, Daniel scored 82, 61,
86, and 82 in his English quizzes. If his lowest score was to be
dropped, which of the following would increase. Is it A) mean, B) median, or C)
mode?
In order to answer this question,
we can calculate the mean, median, and mode before the score was dropped and after
the score was dropped. In order to calculate the mean from
a data set, we firstly add all the values, in this case, 82, 61, 86, and 82. We then divide this total by the
number of values we have, in this case, four. The total, or sum, of Daniel’s
scores is 311. Dividing this by four gives us a
mean of 77.75.
Daniel’s lowest score is being
dropped. This is 61. In order to calculate the mean
after this has been dropped, we add 82, 86, and 82 and then divide by three. This is equal to 84.6
recurring. When Daniel’s lowest score is
dropped, his mean has increased from 77.75 to 84.6 recurring. This suggests that mean is the
correct answer. It is, however, worth checking
whether the median and mode have increased or decreased.
In order to work out the median, we
order the values from smallest to largest or largest to smallest. We then find the middle number. As there are an even number of
values, there will be two middle numbers. As these are both equal to 82, the
median before the lowest score was dropped is 82. After the lowest score is dropped,
Daniel has values of 82, 82, and 86. Once again, the median is equal to
82. Daniel’s median score has,
therefore, not increased.
The mode is the most frequently
occurring value. In both cases, before and after the
lowest score was dropped, the mode was 82. This is because the score 82
occurred more often than any other score. We can, therefore, conclude that
when Daniel’s lowest score was dropped, the median and mode remain the same, but the
mean increased.
Our final question will look at
creating a data set given its range, median, and modes.
A set of four numbers has a range
of seven, a median of 13, and a mode of 16. Given that the highest number is
also the mode, what are the four numbers?
We’re told that there are four
numbers in a data set. Let’s consider them in ascending
order from left to right. We are told that the mode is 16 and
that the highest number is also the mode. As the mode is the most frequently
occurring or most common number, we must have two 16s in the last two boxes.
We are told that the median is
equal to 13. As the median is the middle value,
and there are two middle numbers, each of these must be equidistant from 13. The two numbers in the middle of
our data set are 10 and 16 as these have a median of 13. We are told that the range is equal
to seven. And this is the difference between
the largest and smallest number. 16 minus seven is equal to
nine. Therefore, the smallest number is
nine. The set of four numbers is nine,
10, 16, and 16.
Let’s now look at some of the key
points from this video. We can calculate the mean, median,
mode, and range from a data set. A data set can only have one mean
and one median. There can, however, be one mode,
more than one mode, or no mode at all. There are many different kinds of
mean, but the one that we have used, which is most common, is called the arithmetic
mean. In this video, we have calculated
averages from a data set. We can also do this from discrete
and grouped frequency tables.
