Download 1. Data Presentation and Summary Statistics

Preparatory Mathematics: Collecting and Presenting Data (2015)
1. Data Presentation and Summary Statistics
Section A: Working with Data
An important part of statistical work is presenting data in a clear way from
which conclusions can be drawn.
Example: Each of 40 people is asked how many episodes of “The Sopranos”
they had seen. (40 is the sample size which we will discuss later.)
Answers:
0, 2, 1, 7, 9, 3, 0, 1, 1, 1, 0, 4, 1, 5, 1, 9, 3, 2, 0, 3, 1, 6, 1,
0, 2, 5, 7, 1, 3, 4, 1, 0, 2, 7, 1, 3, 0, 1, 3, 1
The answers are given as a list, but this is not the most helpful way to present
the information.
The “No. of Sopranos Episodes” varies from person to person and is known as
the variable. It is convenient to denote a variable by a letter such as X. Each
different value of the variable occurs with a particular frequency, i.e. the
number of people in each category. The frequency is always denoted by the
letter f (small f).
We could present the information in a Frequency Table.
X
0
1
2
3
4 or more
f
7
13
4
6
10
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Discrete Data
Data is said to be discrete when it can only take a certain set of distinct values.
Examples of Discrete Data:
 The number of children in a family: 0, 1, 2, etc.
 Hourly pay (in euro and cent) in a certain job where the minimum pay is
€8.50 per hour: 8.50, 8.70, 8.95, etc.
We will consider how to present discrete data clearly.
Example: The number of defective items on a production line on 20 successive
days is given below.
12 14 9 13 11 15 10 12 14 13
12 11 13 12 15 9 14 12 13 10
When the figures are presented in this way, it is difficult to make much sense
of them. A frequency table makes clearer how the values in the set of data are
spread out.
Frequency Tables
In the example above, the variable is the number of defective items during a
day. Let X stand for the variable, i.e. X = the number of defective items on a
day.
The number of times each value of the variable occurs is the frequency of the
value, denoted by f.
A frequency table has two columns labelled X and f.
For discrete data we proceed as follows:
1.
Find the lowest and highest values in the set
of data. In our example these are 9 and 15.
2.
Draw the table with two columns, one each
for the variable and the frequency. Label
each column clearly.
3.
Enter each value of the variable in the X
column from the lowest to the highest. Enter
the number of times each value occurs in the
f column opposite the appropriate value.
X (Defective Items)
f (Number of Days)
9
10
11
12
13
14
15
The frequency table for our example is shown opposite.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Section B: Presenting Data
Histograms
A Histogram displays the data from a frequency table graphically. Some people
prefer tables and others pictures!
Put the values of the variable on the horizontal axis (or x-axis) and measure the
frequency on the vertical axis. Draw a rectangle above each value with the
height of the rectangle indicating how often the corresponding value occurs.
Frequency (f) iin
Days
Defective items per Day
6
5
4
3
2
1
0
9
10
11
12
13
14
15
Defective Items in a Day(X)
Some important points when drawing graphs:
1. Your graph must have a title.
2. You must label each axis.
3. You must put units on each axis.
4. If possible indicate where the data comes from in a footnote.
From a Histogram, it is easy to read off which value of the variable occurs most
often. It is simply the value corresponding to the tallest rectangle. This value is
called the mode of the data set. In the above example the mode is 12
defective items – this occurred on more days than any other value.
The mode of a data set is an example of a summary statistic. It is a single
number extracted from the data which can tell us a lot about the data as a
whole. We will see more summary statistics as we go.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Continuous Data and Classes
Data is continuous if it can take any value in some range. If you have a lot of
discrete data of values which are very close to each other then that is usually
seen as continuous data too.
Examples of Continuous Data:
1. The weights of components in a manufacturing process.
2. the length of time it takes a component to break down.
3. The percentage returns on different investments.
For continuous data or discrete data with a large number of values, it is
necessary to group the values in some way before drawing up a frequency
table. This gives us a grouped frequency table.
Note: We have already seen one grouped frequency table. In our very first
example (on Sopranos Episodes) we grouped the values 4, 5, 6, and so on into a
single entry in the frequency table: “4 or more”
Example: The salaries (in €000’s) of 50 employees in a large corporation are
given below.
10.2
21.2
23.49
43.78
42.23
35.7
10.2
16.61
25.55
62.21
12.1
33.76
17.12
37.76
68.37
19.2
34.12
49.16
18.81
11.2
17.5
41.65
55.12
49.12
27.89
44.21
29.43
39.17
38.71
39.87
15.5
21.0
35.63
33.34
52.3
22.3
34.47
41.25
56.72
41.23
25.6
38.72
37.73
30.0
19.54
28.6
32.12
29.15
47.76
28.87
Question: What type of data are these, continuous or discrete?
A straightforward frequency table would be useless here as most of the values
occur with a frequency of either 1 or 2.
We organise the numbers in the list into groups or classes.
1. Find the lowest and highest values in the data set.
2. Choose a class size so that the number of classes is between 5 and 15. The
size itself of each class should be convenient.
3. It is vital that classes do not overlap.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
In our example the lowest salary is 10.2 and the highest is 68.37. Classes of
width 10 or 5 would be appropriate. We will take classes of width 10. Our first
class will be salaries from 10 to 20, the second from 20 to 30 and so on (in
€000’s).
The extreme values in a class (such as 10 and 20 in the first class) are known as
class boundaries.
To prevent the classes from overlapping we must decide between two
alternatives:
(a) Include the lower class boundary in each class and exclude the upper
OR
(b) Exclude the lower class boundary from each class and include the upper.
If we do not make this choice, then we will not know whether the number 20
belongs to the first class or the second class. We will have similar problems for
30, 40, 50, and so on.
We will take option (a).
The first class is salaries “from 10 to under 20 (€000’s)”. 10 belongs to this
class but 20 does not.
The second class is salaries “from 20 to under 30 (€000’s)”. And so on.
Grouped Frequency Table:
X (Salaries in €000’s)
From 10 to under 20
From 20 to under 30
From 30 to under 40
From 40 to under 50
From 50 to under 60
From 60 to under 70
Tally
f (No. of Employees)
The Tally column is there to speed up your counting. As you read through the data put a mark
in the relevant Tally box. Put them in groups of 5 like this as you go
1111
so that each fifth mark is a horizontal line. Why Tally first? – it means you only read ONCE
through the data and ensures you neither miss a value or count it twice.
Now add up your Tally marks to get your frequency column.
Note: A more mathematical notation for “From 10 to under 20” that you might
see is “10  X <20”. We will stick to the english version!
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Histograms for Grouped Data
Drawing a Histogram for grouped data is much the same as it was for simple
discrete data. Again the variable is plotted on the x-axis with each frequency
represented by a rectangle of the correct height. However, this time we have a
rectangle for each class in the frequency table as opposed to a rectangle for
each individual value of the variable.
The Histogram for the salary data is below.
Frequency (f)
Company Salaries
16
14
12
10
8
6
4
2
0
from 10 to
under 20
from 20 to
under 30
from 30 to
under 40
from 40 to
under 50
from 50 to
under 60
from 60 to
under 70
Salaries in €000's (X)
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Practice Example: A group of 25 people are asked for their weight to the
nearest lb. Here are the answers.
145, 143, 161, 156, 159, 159, 154, 153, 167, 155, 151, 146, 148, 160, 134, 143,
155, 157, 142, 171, 146, 163, 161, 153, 172
A Grouped Frequency Table for these data is given below.
X (Weights in lbs)
From 130 to under 140
From 140 to under 150
From 150 to under 160
From 160 to under 170
From 170 to under 180
Tally
f (frequency)
0
0
0
0
0
Histogram:
Frequency (f)
Weights
12
10
8
6
4
2
0
From 130 to From 140 to From 150 to From 160 to From 170 to
under 140
under 150
under 160
under 170
under 180
Weights in lbs (X)
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Section C: Summary Statistics
“Population” and “Sample”
The word “population” refers to a complete data set.
Examples:
 If we are dealing with rolls of a die, the population is the results of every
roll of the die ever done.
 If we are dealing with the heights of men in Ireland, the population is the
heights of every man in the country.
 If we are % carbon content in a piece of steel from a production run, the
population might be all of the pieces produced on one run. It may also be
all pieces which might be produced by that production process into the
future.
PROBLEM: It is often difficult or impossible to work with the population as a
whole. In the last example, we will never know all of the % carbon content
figures on all steel pieces.
We select a sample from the population and work with that instead. In
manufacturing, samples from a production run are usually tested for quality
control purposes. This sampling and testing may be destructive and/or time
consuming, so you obviously don’t want to test everything!
Examples of Samples:
 Roll a die 250 times and record the results.
 Select 1000 men at random and measure their heights.
 Select 10 steel pieces at random from a day’s production run and test their
% carbon content.
The sample should represent the whole population.
Examples of Bad Sampling:
 The population is the voting intentions of every voter in Ireland.
The sample (opinion poll) is the voting intentions of 100 voters attending
the Labour Party Conference.

The population is the heights of all the men in Ireland.
The sample is the heights of 1000 male Gardaí.

The sample is the first 10 steel pieces in the day’s production run.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Statistics and Parameters
A parameter is a number that you
calculate using the entire population.
Example, calculating the mode from the
whole population would give us a
parameter.
A parameter is constant for the
population.
A statistic is a number that we
calculate using only a sample.
Parameters are difficult or more usually
impossible to calculate!
Statistics can always be calculated.
A statistic can vary from one sample to
another and is not constant for the
population.
A statistic is used to estimate a parameter. The obvious question is “when is a
statistic a good approximation for the parameter i.e. how big does my sample
need to be?” We will look at this problem next semester – for now, we will
simply calculate some statistics.
A summary statistic is a statistic that sums up the data in a sample – it tells you
something about the data as a whole.
There are two types of summary statistic:
1. Measures of Location (also called Central Tendency)
2. Measures of Dispersion
Measures of Location
An average is a point within a group of data which is central to the group, and
around which the other values are distributed. It is therefore a measure of
central tendency – a measure which starts to summarise the data by fixing one
point as the centre. The position of the central item fixes the location of the
distribution and averages are therefore sometimes called measures of location.
There are three measures of location that we will discuss: the mode, the
median, and the mean.

Mode – most frequently occurring value

Median – “middle” value when all the numbers are arranged in order. (It is
greater than half of the values in the data set and less than half the values
in the data set.)

Mean – “average” value in the familiar sense. The mean is the arithmetic
average: add up the values and divide by however many of them there are.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
The mode, the median, and the mean all give an indication of where the data
is situated. The idea in each case is to pick one number that is representative
of the data set as a whole.
Example: For the following set of numbers, calculate the mode, the median,
and the mean.
12, 11, 14, 7, 0, 1, 11, 8, 11, 2, 3
Firstly, write the numbers in ascending order:
0, 1, 2, 3, 7, 8, 11, 11, 11, 12, 14

Mode = 11
(the number that occurs most often – writing the
numbers in ascending order groups the 11’s together
making it easier to identify the mode)

Median = 8
(the middle number – only when the numbers are
written in ascending order can we see which one is in the
middle)

To calculate the mean we add up the numbers and divide by how many
numbers we have:
Add up the numbers: 12+11+14+7+0+1+11+8+11+2+3=80
There are 11 numbers in the list. This is n, the sample size.
Mean =
80
= 7.27
11
Formula for the mean:
Note: The symbol

(to two decimal places)
x
x
n
means to “add up”.
Note: The symbol x is always used to denote the mean of a sample of
numbers.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Example
Find the mean for the salary data given before:
The salaries (in €000’s) of 50 employees in a large corporation are given below.
10.2
21.2
23.49
43.78
42.23
35.7
10.2
16.61
25.55
62.21
12.1
33.76
17.12
37.76
68.37
19.2
34.12
49.16
18.81
11.2
17.5
41.65
55.12
49.12
27.89
44.21
29.43
39.17
38.71
39.87
15.5
21.0
35.63
33.34
52.3
22.3
34.47
41.25
56.72
41.23
25.6
38.72
37.73
30.0
19.54
28.6
32.12
29.15
47.76
28.87
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Calculating the Mode from a Histogram
The mode of a data set can easily be found using a Histogram of the data. The
mode is just the value (or class of values) with the tallest rectangle in the
Histogram. When it is a class of values it is called the Modal Class.
Example: Below is the histogram for the Defective Items data. It is easy to pick
out the mode of the data set.
Frequency (f)
Defective Items Data
6
4
2
0
9
10
11
12
13
14
15
Num ber of Defective Item s (X)
Mode =
What is the modal Class for the Salaries data?
What is the modal Class for the weights data?
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Estimating the Median for larger data sets
To estimate the median a set of numbers arrange all of the numbers in
ascending order and pick out the number which is half way along the list.
To estimate the median of a set of numbers where there are a large number of
data it is easiest to arrange the data in ascending order by putting it into a
spreadsheet and using the sort function ( see the lab notes for how to do this).
Example: Recall that a group of 25 people are asked for their weight to the
nearest lb. Here are the answers.
145, 143, 161, 156, 159, 159, 154, 153, 167, 155, 151, 146, 148, 160, 134, 143,
155, 157, 142, 171, 146, 163, 161, 153, 172
Estimate the median
Solution:
Median ≈
Example
Find the median for the salary data given before:
The salaries (in €000’s) of 50 employees in a large corporation are given below.
10.2
21.2
23.49
43.78
42.23
35.7
10.2
16.61
25.55
62.21
12.1
33.76
17.12
37.76
68.37
19.2
34.12
49.16
18.81
11.2
17.5
41.65
55.12
49.12
27.89
44.21
29.43
39.17
38.71
39.87
15.5
21.0
35.63
33.34
52.3
22.3
34.47
41.25
56.72
41.23
25.6
38.72
37.73
30.0
19.54
28.6
32.12
29.15
47.76
28.87
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Solution:
Enter the data into a spreadsheet and using the sort
function rearrange in ascending order as follows:
Median =
List
Position on
list
Reverse
Position on list
10.2
10.2
11.2
12.1
15.5
16.61
17.12
17.5
18.81
19.2
19.54
21
21.2
22.3
23.49
25.55
25.6
27.89
28.6
28.87
29.15
29.43
30
32.12
33.34
33.76
34.12
34.47
35.63
35.7
37.73
37.76
38.71
38.72
39.17
39.87
41.23
41.25
41.65
42.23
43.78
44.21
47.76
49.12
49.16
52.3
55.12
56.72
62.21
68.37
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Advantages/Disadvantages of the Three Measures of Location
It should be noted that for a reasonably large set of real data, the three
measures of location will be more or less the same.
Mean
 Advantages
Takes all the numbers into account
Easy to calculate
 Disadvantages
Affected by large values
Can be a number not in the actual
data set
Familiar
Useful Mathematical properties

When to use the mean - if all of the values in a data set are roughly equal,
the mean is the best number to use as a summary statistic.
Mode

Advantages
Simple to calculate
Unaffected by large values
 Disadvantages
May be unrepresentative of the whole
data set
Might not be unique. For example,
consider the list of numbers 1, 2, 2, 2,
4, 4, 7, 7, 7, 9. Both 2 and 7 are modes
for that set!
Useful for non-numerical data as
well

When to use the mode - when nearly all the values in the data set are the
same or for a small data set.
Median
 Advantages
Unaffected by large values
 Disadvantages
May not be representative of the whole
data set
Easy to calculate

When to use the median - when the data set has a few very large or very
small numbers.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Skewed Data( http://www.mathsisfun.com/data/skewness.html)
Data can be "skewed", meaning it tends to have a long tail on one side or the
other:
Negative Skew
No Skew
Positive Skew
Negative Skew?
Why is it called negative skew?
Because the long "tail" is on
the negative side of the peak.
People sometimes say it is
"skewed to the left"
(the long tail is on the left hand side).
The mean is also on the left of the peak.
Not Skewed
A Normal Distribution is not skewed.
It is perfectly symmetrical.
And the Mean is exactly at the peak
Positive Skew
And positive skew is when
the long tail is on the
positive side of the peak,
and some people say it is
"skewed to the right".
The mean is on the right of the peak value.
Calculating Skewness
"Skewness" (the amount of skew) can be calculated, for example you could use
the SKEW() function in Excel. Some other resources:
http://www.statisticshowto.com/skewed-distribution/
Positive Skew
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And positive skew is when the long tail is on
the positive side of the peak, and some people
Preparatory Mathematics: Collecting and Presenting Data (2015)
Measures of Dispersion
While the mode, median, and mean help us to summarise a set of data, they
tell us nothing about how spread out the values in the set are.
Example: Look at the following two sets of numbers
 22, 22, 23, 24, 25, 27, 27
 18, 20, 21, 24, 27, 28, 30
For both, the mean = 24, but the second is obviously more spread out.
We will look at two ways of describing the amount of spread in a set of data:
1. Range
2. Standard Deviation
Range
The range is just the difference between the lowest and highest numbers in the
data set.
Example:
22, 22, 23, 24, 25, 27, 27
Range = 27-22 = 5

Advantages
1. Easy to calculate

Disadvantages
1. Only use two numbers from the set
2. One outlier can skew the range as a useful measure if all the other
values are close together.
The range, as a measure of dispersion, is not extensively used. It has some
usefulness for certain “quality control” systems, where values (e.g.
temperature) cannot be allowed to stray outside a certain range.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Standard Deviation
The standard deviation is the most important and widely used measure of
dispersion. It measures the average deviation of the numbers in a data set from
the mean of the set.
Example: Calculate the standard deviation of the sample data set
177, 180, 181, 185, 187
1. First we calculate the mean
Mean = x =
2. Next subtract the mean away from each number in the original set.
177 −
=
185 −
=
180 −
=
187 −
=
181 −
=
3. Now square each of these “deviations” from the mean.
4. Next calculate the average (mean) of these squared deviations. (This
average is known as the sample variance and is denoted by s2.)
s2 =
5. Finally, to compensate for having squared all of the original deviations, take
the square root of the answer from part 4. This is the standard deviation of
the set of numbers.
Standard deviation s =
=
Note: For a small sample such as this we should really use the “sample
standard deviation” formula, which in step 4 above would divide by n – 1 = 4
rather than n = 5. or larger samples, there is no great difference between using
n and n – 1, so we will use n and not worry about it.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Example:
Find the mean and the standard deviation of :
3
3.2
4.1
4.5
5.5
Solution:
Steps:
1. First we calculate the mean
2. Next subtract the mean away from each number in the original set.
3. Now square each of these “deviations” from the mean.
4. Next calculate the average (mean) of these squared deviations s2 =
5. Finally to get the standard deviation of the set of numbers, take the square
root of the answer from part 4. This is
Formula for Standard Deviation
s
e x  x j
x
3
3.2
4.1
4.5
5.5
2
(again, use n – 1 if n is small)
n
mean
x-mean
d x  x i
x =
x =
(x-mean)^2
and
s
2

e x  x j
n
2
=
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Formula for Standard Deviation
e x  x j
s
2
(again, use n – 1 if n is small)
n
Note: This is the standard deviation formula when frequencies are not known.
Example: The number of days with an average temperature below 6oC is shown
below. Calculate the mean and standard deviation for this data.
Jan
18
Feb
12
Mar
11
Apr
6
May
9
Jun
4
Jul
2
Aug
0
Sep
2
Oct
6
Nov
12
Dec
17
Solution:
We have 12 numbers here: 18,12,11,6 ,9, 4, 2, 0, 2, 6, 12, 17 so here n = 12
Standard Deviation s 
e x  x j
2
n
x
x
18
12
11
6
9
4
2
0
2
6
12
17
2
d x  x i
x =
x =
dx  xi
xx
and
s
e x  x j
n
2

2
=
This is method is fine for small data sets and in the lab with the use of an
excel spreadsheet we can use it for large data sets also.
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Six sigma picture:
http://www.google.co.uk/imgres?imgurl=http://nursingplanet.com/biostatistics/normal_c
urve.jpg&imgrefurl=http://nursingplanet.com/biostatistics/normal_distribution_and_prob
ability.html&h=331&w=784&sz=20&tbnid=c7hWm9OjFIPjAM:&tbnh=52&tbnw=124
&prev=/search%3Fq%3Dstandard%2BDeviation%2B6%2Bsigma%2Bpictures%26tbm%3Disch%26tbo%3Du&zoom=1&q=standard+Deviati
on+-6+sigma+pictures&docid=0WVRBIa8D5MITM&hl=en&sa=X&ei=cSCYT4XyKIDhQehzdn1BQ&ved=0CDAQ9QEwAg&dur=8406
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Preparatory Mathematics: Collecting and Presenting Data (2015)
Problem Sheet
Data Presentation and Summary Statistics
1. Find the mean, median and mode of the following sets of data:
(a) 5, 7, 10, 19,
7, 8, 16, 11,
9, 12, 14.
(b) 127
121
122
138
123
129
124
128
132
124.
2. Find the range and standard deviation for each of the data sets in question
1.
3. A firm has a team of 5 maintenance technicians who service machines leased
to customers. The following frequency distribution records the number of days
when some or all of the technicians were out on service calls during a 40 day
period:
Number of Technicians
Number of Days
0
3
1
10
2
8
3
7
4
6
5
6
Plot a histogram for this data.
What is the mode for these data? Find the mean number of technicians out on
call per day.
By writing down a list of these numbers calculate what is the median for these
data? Comment on the results for the mean and median.
Page 22 of 25
Preparatory Mathematics: Collecting and Presenting Data (2015)
Back
count
Couint
1
2
11
10
x
5
7
meanx
(x-meanx)
(x-meanx)^2 Ps1(a)
10.5455
-5.54545455
30.75206612 Mean =
10.5455
-3.54545455
12.57024793
10.5455
-3.54545455
10.54545455
Standard
12.57024793 Deviation =
10.5455
-2.54545455
6.479338843
10.5455
-1.54545455
2.388429752
10.5455
-0.54545455
0.297520661 n=
10.5455
0.454545455
0.20661157
10.5455
1.454545455
2.115702479
10.5455
1.454545455
2.115702479 Mode = 7 and 12
10.5455
5.454545455
29.75206612 Median =10
10.5455
8.454545455
71.47933884 Range =19-5 =
7
3
9
4
8
5
7
6
6
7
5
8
4
9
3
10
2
11
1
8
9
10
11
12
12
16
19
3.939627033
11
14
116
is the sum
of the x
170.7272727
Is Sum of (x-meanx)^2
Page 23 of 25
Preparatory Mathematics: Collecting and Presenting Data (2015)
Back
Couint count
meanx
(x-meanx)
(x-meanx)^2
PS 1(b)
126.8
-5.8
33.64
Mean =
126.8
-4.8
23.04
126.8
-3.8
14.44
1
10
x
121
2
9
122
3
8
4
7
124
126.8
-2.8
7.84
5
6
124
126.8
-2.8
7.84
6
5
127
126.8
0.2
0.04
7
4
128
126.8
1.2
1.44
8
3
129
126.8
2.2
4.84
9
2
132
126.8
5.2
27.04
126.8
11.2
125.44
123
138
10
1
126.8
Standard
Deviation =
4.955804677
n=
10
Mode = 124
Median
=(124+127)/2 =
125.5
Range =138-121 =
17
1268
is the sum
of the x
245.6
Is Sum of (x-meanx)^2
Page 24 of 25
Preparatory Mathematics: Collecting and Presenting Data (2015)
x
18
meanx
(x-meanx)
(x-meanx)^2
8.25
9.75
95.0625
12
8.25
3.75
14.0625
11
8.25
2.75
7.5625
6
8.25
-2.25
5.0625
9
8.25
0.75
0.5625
4
8.25
-4.25
18.0625
2
8.25
-6.25
39.0625
0
8.25
-8.25
68.0625
2
8.25
-6.25
39.0625
6
8.25
-2.25
5.0625
12
8.25
3.75
14.0625
17
8.25
8.75
76.5625
99
is the sum of the x
Temp Notes Eg
Mean =
8.25
Standard Deviation =
5.643949563
n=
12
382.25
Is Sum of (x-meanx)^2
Page 25 of 25