What Is The Median Of The Data Set? 9, 3, 10, 12, 4, 5, 12, 2 Enter Your Answer In The Box.

Unit v Department two : Mean, Median, Mode and Range

The hateful, median and way are types of average.
The range gives a measure out of the spread of a fix of data.

This section revises how to calculate these measures for a simple set of information.
It then goes on to await at how the measures can be calculated for a table of data.

Computing the Hateful, Median, Mode and Range for simple data

The table beneath shows how to summate the mean, median, mode and range for two sets of data.
Fix A contains the numbers ii, 2, 3, v, 5, vii, 8 and Set B contains the numbers ii, 3, iii, four, 6, 7.

Measure out

Set A
2, 2, iii, 5, five, 7, viii

Set B
2, three, 3, 4, vi, 7

The Mean
To find the hateful, you lot
demand to add up all the
data, and so divide
this total past the number
of values in the information.

Calculation the numbers up gives:
two + 2 + 3 + 5 + 5 + 7 + 8 = 32

There are 7 values, and so you lot divide
the total past 7:32 � 7 = four.57...

Then the hateful is 4.57 (2 d.p.)

Adding the numbers upward gives:
2 + iii + 3 + 4 + 6 + seven = 25

In that location are 6 values, and then you lot divide
the total past 6:25 � 6 = 4.166...

Then the mean is 4.17 (ii d.p.)

The Median
To detect the median, you
need to put the values
in order, and then notice the
heart value. If at that place are
two values in the eye
then you find the mean
of these 2 values.

The numbers in club:
2 , 2 , three , (v) , 5 , vii , 8

The centre value is marked in
brackets, and information technology is 5.

And so the median is v

The numbers in gild:
2 , iii , (iii , 4) , 6 , seven

This fourth dimension there are two values in
the center. They take been put
in brackets. The median is found
by computing the hateful of these
two values:(3 + four) � ii = 3.v

So the median is 3.5

The Mode
The way is the value
which appears the nigh
ofttimes in the data. It is
possible to have more than
than one mode if there
is more than than 1 value
which appears the most.

The data values:
2 , 2 , 3 , 5 , 5 , 7 , 8

The values which appear most
oftentimes are ii and 5. They both
announced more time than any
of the other data values.

And then the modes are 2 and v

The data values:
two , three , 3 , iv , half-dozen , 7

This fourth dimension there is only one value
which appears almost often - the
number iii. It appears more times
than whatsoever of the other data values.

And so the mode is 3

The Range
To notice the range, you
first need to find the
lowest and highest values
in the data. The range is
found by subtracting the
lowest value from the
highest value.

The data values:
2 , 2 , 3 , 5 , 5 , 7 , 8

The lowest value is two and the
highest value is 8. Subtracting
the lowest from the highest
gives:viii - 2 = 6

And then the range is 6

The data values:
2 , 3 , 3 , 4 , 6 , seven

The lowest value is 2 and the
highest value is 7. Subtracting
the lowest from the highest
gives:7 - 2 = 5

And then the range is 5

Practice Question (for simple information)
Piece of work out the mean, median, way and range for the simple information set below,
so click on the button marked to see whether you are right.

A data ready contains these 12 values: three, v, 9, 4, 5, eleven, 10, 5, vii, 7, 8, ten

(a) What is the mean?

(b) What is the median?

(c) What is the mode?

(d) What is the range?

Calculating the Hateful, Median, Fashion and Range for a tabular array of data

Sometimes nosotros are given the data in a table. The methods for computing mean, median, way
and range are exactly the aforementioned, only nosotros need to retrieve carefully about how we acquit them out.
In this department we volition employ 1 set of data in a table and calculate each measure in turn.

Case
A die was rolled xx times. On each roll the die shows a value from i to 6.
The results have been recorded in the table beneath:

Value	Frequency
i	three
2	v
3	2
iv	iv
five	3
six	three

The frequency is the number of times each value occured.
For example, the value 1 was rolled three times, the value two was rolled 5 times and and so on...

When nosotros want to think nigh calculating the measures for this data gear up, it can be helpful
to recollect about what the numbers would look similar if we wrote them out in a listing:
1, 1, 1, 2, 2, 2, 2, 2, three, 3, 4, iv, four, four, 5, five, 5, 6, 6, 6

We could just calculate the mean, median, style and range from this list of data, using
the methods described in the beginning function of this section. The problem is that if there were
hundreds of values in the table then information technology would have a long time to write out the listing of data
and even longer to do the calculations. It would exist meliorate if nosotros could piece of work directly from
the table to summate the measures. The method for doing this is shown beneath.

Finding the hateful from a tabular array of data

Value	Frequency
1	3
2	5
three	two
iv	4
5	3
vi	three

We know that if we write the case data in a list it looks like this:
one, 1, 1, 2, 2, 2, 2, two, 3, iii, 4, 4, four, 4, five, five, 5, half dozen, 6, 6

Normally nosotros would add up the data and carve up the full by the number of values:
The full is 1+one+ane + 2+2+ii+2+ii + 3+iii + four+4+4+4 + 5+5+v + 6+6+6 = 68
The number of values is xx, so the mean is 68 � 20 = 3.iv

We could have plant these figures more easily! To become the total, we have added
up 3 lots of "1", 5 lots of "two", 2 lots of "3", iv lots of "iv", 3 lot of "5" and 3 lots of "6".
This is the same calculation equally 3�one + 5�ii + two�3 + 4�iv + 3�5 + 3�6 = 68.
We have multiplied each value by its frequency and added upwardly the results to become the total
of all the values. Nosotros can likewise go the "number of values" more easily by simply calculation
up all the frequencies: iii + 5 + two + 4 + 3 + 3 = xx

And so how do we do this in a table?

Firstly, yous need to add an extra cavalcade in the table:
This is where you multiply each value by its frequency. For example,
the value v has a frequency of 3, so we multiply 5 by 3 to become 15.

Secondly, you need to calculate ii of import totals:
(i) add up the values in the frequency column to find out the
number of data values. In this instance there are 20 values.
(2) add up the values in the value � frequency column to observe
out the total of all the data values. In this instance the total is 68.

Finally, you need to summate the mean:
To practise this, divide the total of all the information values by the number of
data values. In this example you need to split 68 by 20, giving 3.four.

Value	Frequency	Value � Frequency
1	three	i � 3 = 3
2	five	two � 5 = x
3	2	3 � two = vi
4	4	four � 4 = 16
5	3	5 � iii = 15
6	three	6 � three = 18
Totals	xx	68

This method of computing the mean for a table of information is exactly the same every bit the i used with a list of data.
Nosotros have nonetheless added upwardly all the values and divided by the number of values, but this way is a bit more efficient!

Finding the median from a table of information

Value	Frequency
1	3
2	5
3	ii
4	iv
v	3
vi	3

We know that there are xx data values in our tabular array. If you imagine the 20 values
written out, at that place would be two values in the middle. These would be the tenth and
11th values, and the median would be the hateful of these ii "middle values".

From the list below we tin see that the "heart values" are three and 4:
1, ane, 1, two, two, ii, 2, 2, 3, 3, 4, 4, iv, 4, 5, 5, v, half dozen, six, 6
The median would therefore be (three+iv)�ii = 3.v

So how do we do this from a table?
Because there are twenty values, we know that we need to find the mean of the 10th and
11th values. To discover these values nosotros demand to count through the table until we get to them.

Await at the tabular array. The value "ane" has a frequency of 3, then the outset three values in the tabular array are "ane"s.
The value "2" has a frequency of 5, and then the next five values are all "2"s. This takes united states of america up to the 8th value.
The adjacent 2 values are "3"s, which takes us up to the tenth position in the data, then the tenth value must be a "iii".
The adjacent 4 values are "four"s, so the 11th value must be a "4".

We can at present meet that the 10th and 11th values are a "3" and a "4", so the median is 3.5.

Finding the fashion and range from a table of data

Value	Frequency
ane	3
two	5
three	2
4	4
5	3
half dozen	three

Finding the mode is much easier from a table, because the frequency column
tells united states how many times each value occured. We tin notice the value which
occured the most often by looking for the value with the highest frequency.
In this case we can see that the value with the highest frequency is "ii".
The mode of this prepare of data is therefore 2

Finding the range is besides easy from a tabular array. To find the highest and lowest data
values, you simply look for the highest and lowest values in the values column.
In this case the everyman value is "one" and the highest value is "6", and 6 - 1 = 5.
The range of this set of data is therefore 5

Do Question (for data in a table)
Work out the each of the measures, then click on the push marked

to come across whether you lot are correct.

Value	Frequency	Value � Frequency
0	3
i	7
two	x
3	8
4	i
v	1
Totals

You can fill up in the boxes to aid yous with your working simply they volition not exist marked.

30 couples were asked how many children they have.
The results are shown in the table on the left.

(a) What is the mean number of children?

(b) What is the median number of children?

(c) What is the way?

(d) What is the range?

Exercises

Piece of work out the answers to the questions below and make full in the boxes. Click on the

button to find out whether you take answered correctly. If you are right then

will announced and you should move on to the next question. If

appears and so your answer is wrong. Click on

to clear your original answer and have another go. If you can't work out the correct answer and so click on

to meet the answer. Yous may discover it helpful to have pencil and paper to practice workings for these questions. Question 1
Calculate the mean, median, mode and range for each set of data beneath:

(a)	iii, half-dozen, 3, 7, 4, 3, 9	Mean =
		Median =
		Manner =
		Range =
(b)	xi, ten, 12, 12, 9, x, fourteen, 12, 9	Mean =
		Median =
		Mode =
		Range =
(c)	2, nine, 7, three, v, 5, 6, five, four, ix	Hateful =
		Median =
		Style =
		Range =

You accept at present completed Unit v Department 2

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Produced by A.J. Reynolds July 2003 >