This tutorial explain the measure of central tendency (Mean, Median and Mode in detail with suitable working examples pictures. The tutorial also teach the excel commands for calculation of Mean, Median and Mode.
This tutorial explain the measure of central tendency (Mean, Median and Mode in detail with suitable working examples pictures. The tutorial also teach the excel commands for calculation of Mean, Median and Mode.
Senior Software Developer and Lead Trainer Alejandro Lujan explains pattern matching, a very powerful and elegant feature of Scala, using a series of examples.
Learn more about this topic and find more presentation on Scala at:
In his latest Typesafe tutorial video, Alejandro Lujan explains for expressions in Scala, and provides an example of them in action.
For expressions are a very useful construct that can simplify manipulation of collections and several other data structures. They can be used in place of nested for loops, or to replace calls to map and flatMap in non-collection structures.
Learn more
This presentation provides an overview on Value Classes in Scala, which is explained in the video on the last slide by Alejandro Lujan. He explains why you would want to use them, outlines the restrictions that are associated with them, and shows examples of how you would use them. Value classes are a mechanism that Scala provides to create a certain type of wrapper classes that provide memory and performance optimizations. In this video, we show a use case for Tiny Types with Value classes.
Alejandro Lujan introduces us to String Interpolation, a feature of Scala that allows us to have placeholders inside of string definitions, and explains why you would want to use them. Video included!
This presentation explores the benefits of functional programming, especially with respect to reliability. It presents a sample of types that allow many program invariants to be enforced by compilers. We also discuss the industrial adoption of functional programming, and conclude with a live coding demo in Scala.
Senior Software Developer Alejandro Lujan discusses the collections API in Scala, and provides some insight into what it can do with with some examples.
In this video, senior software developer Alejandro Lujan explores the elements of Scala's language that allow you to write clean and powerful code in a more brief manner.
Did you miss Scala Days 2015 in San Francisco? Have no fear! BoldRadius was there and we've compiled the best of the best! Here are the highlights of a great conference.
As a full-time Scala developer, I often find myself talking about Scala and functional programming in different kinds of situations, ranging from meeting a friend working in J2EE, Ruby or C++, to dedicated Scala Meetups aiming to promote deeper understanding of the language. However, something occurred to me lately. By hanging out with people who have some Scala knowledge or experience, I am somewhat holding on to a safe place. By presenting only to people who are curious about Scala, I'm preaching to the converted.
To make a long story short, I recently made an attempt at getting out of my comfort zone by presenting about how making the transition from Java to Scala makes total sense (from Java developer point of view). The presentation went through proof-hearing of approximately 60 experienced Java programmers (with almost no prior Scala knowledge) gathered in one room for a Lunch & Learn. Here are my slides.
Punishment Driven Development #agileinthecityLouise Elliott
What is the first thing we do when a major issue occurs in a live system? Sort it out of course. Then we start the hunt for the person to blame so that they can suffer the appropriate punishment. What do we do if a person is being awkward in the team and won’t agree to our ways of doing things? Ostracise them of course, and see how long it is until they hand in their notice – problem solved.
This highly interactive talk delves into why humans have this tendency to blame and punish. It looks at real examples of punishment within the software world and the results which were achieved. These stories not only cover managers punishing team members but also punishment within teams and self-punishment. We are all guilty of some of the behaviours discussed.
This is aimed at everyone involved in software development. It covers:
• Why we tend to blame and punish others.
• The impact of self-blame.
• The unintended (but predictable) results from punishment.
• The alternatives to punishment, which get real results.
SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docxanhlodge
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
μ = 7
Step 2. Then for each number: subtract the Mean and square the result
This is t.
SAMPLING MEANDEFINITIONThe term sampling mean is a stati.docxagnesdcarey33086
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
μ = 7
Step 2. Then for each number: subtract the Mean and square the result
This is t.
SAMPLING MEAN DEFINITION The term sampling mean is.docxagnesdcarey33086
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical
distributions. In statistical terms, the sample mean from a group of observations is an
estimate of the population mean . Given a sample of size n, consider n independent random
variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these
variables has the distribution of the population, with mean and standard deviation . The
sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that
it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number
of items taken from a population. For example, if you are measuring American people’s weights,
it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the
weights of every person in the population. The solution is to take a sample of the population, say
1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are
going to analyze. In statistical terminology, it can be defined as the average of the squared
differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
• Determine the mean
• Then for each number: subtract the Mean and square the result
• Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
• Next we need to divide by the number of data points, which is simply done by
multiplying by "1/N":
Statistically it can be stated by the following:
•
• This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, µ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
The mean is:
(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
So:
µ.
SAMPLING MEAN DEFINITION The term sampling mean .docxanhlodge
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical
distributions. In statistical terms, the sample mean from a group of observations is an
estimate of the population mean . Given a sample of size n, consider n independent random
variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these
variables has the distribution of the population, with mean and standard deviation . The
sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that
it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number
of items taken from a population. For example, if you are measuring American people’s weights,
it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the
weights of every person in the population. The solution is to take a sample of the population, say
1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are
going to analyze. In statistical terminology, it can be defined as the average of the squared
differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
• Determine the mean
• Then for each number: subtract the Mean and square the result
• Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
• Next we need to divide by the number of data points, which is simply done by
multiplying by "1/N":
Statistically it can be stated by the following:
•
http://www.statisticshowto.com/find-sample-size-statistics/
http://www.mathsisfun.com/algebra/sigma-notation.html
• This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each bush is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Work out the sample variance
Step 1. Work out the mean
In the formula above, μ (the Greek letter "mu") is the mean of all our values.
For this example, the data points are: 9, 2, 5, 4, 12, 7, 8,.
Measures of Central Tendency, Variability and ShapesScholarsPoint1
The PPT describes the Measures of Central Tendency in detail such as Mean, Median, Mode, Percentile, Quartile, Arthemetic mean. Measures of Variability: Range, Mean Absolute deviation, Standard Deviation, Z-Score, Variance, Coefficient of Variance as well as Measures of Shape such as kurtosis and skewness in the grouped and normal data.
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
1. Empirics of Standard Deviation
In research, there are the different methods of measuring data to be analyzed. The reason for these is to
measure the level of dispersion (Eboh, 2009). Dispersion is the tendency of values of a variable to scatter away from
the mean or midpoint. The data are measured majorly with basic statistical tools such as mean, median and mode. To
arrive at accurate measurement, the use of standard deviation is employed. Standard deviation is a measurement that
is designed to find the disparity between the calculated mean.it is one of the tools for measuring dispersion. To have a
good understanding of these, it is of general interest to give a better light to the following terms (mean, median, mode)
and variance) also their uses.
MEAN
Panneerslvam (2008) defined mean as the ratio between the sum of the observations and the number of the
observation.in his study, he termed it as arithmetic mean. .Eboh (2009) said it is sum of observations divided by the
number of observations. Mathematically, the mean is the arithmetic average of a number of scores. To obtain the
mean, add your scores and divide by the number of scores that you have. Simply put that the mean is the addition of
all the collated data that are to be analyzed, which is then divided by the number of the data to the analyzed.it is
generally stated as
/x = ∑ 𝑥𝑖𝑛
𝑖=1
/n
Where /x is the arithmetic mean; xi, the ith observation; and n, the total number of observations
Example 1.1 determine the arithmetic mean of salaries of the employees s shown in the table 1.1 below
Employees no. 1 2 3 4 5 6 7 8 9
Monthly salary N
,000
20 27 34 56 34 45 20 29 41
Solution-----------The number of observations, n =9
Using the above formula, /x = ∑ 𝑥𝑖𝑛
𝑖=1
/n
20000+27000+34000+56000+34000+45000+20000+29000+41000 = N34000
9
It should be noted that before summing them up, they must be in the same units and also in the same scale. This means
that there can’t different values that ought to be summated, such as having naira and dollars values that are to be
summated, it will be impossible to do so. The summation of these two different scales of measurement won’t be
possible. Consider the following data, which represents the time needed to complete a reading task, as an example.
2. Example 1.2
Times in
miuntes
6 3 5 5 2 7 6 4 3
Total = 43
The mean is the sum of scores divided by the number of scores, mathematically: Mean = ΣX/N = 43/10 = 4.3
PROPERTIES of THE MEAN
The mean has certain properties that are attributed to it (Eboh, 2009). They include
1. It has algebraic property that the sum of the deviations of each observation from the mean will always be zero.it
means that when the mean observation is subtracted from the mean and summed together (which will comprise of
both the positive and negative values), it must result to zero. This is expressed mathematically as thus:
∑ (𝑥𝑖 − 𝑥
)𝑁
𝑖=1 = 0
2. The sum of the squared deviations of each observation from the mean is less than the sum of the squared deviations
about any other number
∑ (𝑥𝑖 − 𝑥
)𝑁
𝑖=1 2= minimum
This means that the when the various values that were computed together to form the mean are being subtracted,
originally when summed up, they will give a zero value. But when squared together after the subtraction from the
mean, the result arrived it the minimum value.
3. When mean is commuted from a grouped data which is a special case, midpoints of each assumed that each of the
interval classes is being assumed. This is illustrated mathematically as this
𝑋−
= ∑ 𝒌
𝒊=𝟏 fi mi =
N
Where fi = number of cases in the ith
category, with f, =N
M1 = midpoint of the ith
category
K = number of categories
This is further expressed as finding the idle point of a grouped data that is expected to be analyzed.an example of a
grouped data is 1950-2950. Such a grouped data has its midpoint has 2,450.the 2,450 is what will be used for mean
analysis.
MEDIAN
According to R.panneerslvam (2008), the median is the score found at the exact middle of the set of values. It refers to
the midpoint in a series of numbers. To find the median the values are arranged in order from smallest to largest. If
there is an odd number of values, the middle value is the median. If there is an even number of values, the average of
the two middle values is the median.
3. Example 1.3: Find the median of 19, 29, 36, 15, and 20
In order: 15, 19, 20, 29, 36 since there are 5 values (odd number), 20 is the median (middle number)
Example 1.4: Find the median of 67, 28, 92, 37, 81, 75
In order: 28, 37, 67, 75, 81, 92 since there are 6 values (even number), we must average those two middle numbers
to get the median value. Average: 67 + 75 = 142 = 71 is the median value
2 2
MODE
The mode of a set of values is the value that occurs most often. A set of values may have more than one mode or no
mode.
Example 1.5: Find the mode of 15, 21, 26, 25, 21, 23, 28, 21
The mode is 21 since it occurs three times and the other values occur only once.
Example 1.6: Find the mode of 12, 15, 18, 26, 15, 9, 12, 27
The modes are 12 and 15 since both occur twice.
Example 1.7: Find the mode of 4, 8, 15, 21, 23
There is no mode since all the values occur the same number of times.
Since there are 3 different measures of centers, it seems reasonable to ask which is best to use. There are advantages
and disadvantages to each of them, depending on the nature of the data set. These are listed below.
Measure Advantages Disadvantages
Mean Easy to Compute
Sample Means tend to Vary Less
Good properties as sample size increases
(more to come on that later)
Sensitive to extreme values (outliers)
Median Resistant to outlying values
Good for skewed data (see below)
Harder to calculate
Less useful than the mean for inference
(more to come on that later)
Mode Easy to compute
Good for qualitative (categorical) data
Not very useful for quantitative data
Skewness
Using the mean, median, and mode together can help to describe the skewness of a data set. A data set is
considered skewed if the values extend more to one side of the distribution than the other. (Schuetter, 2007)
4. VARIANCE
The variance ( S2
) is the average squared deviation from the mean. It is also known as the square of the standard
deviation. Both measures are interchangeable. These means that the standard deviation is the square root of the variance.
The defining formula is
S2
=
∑(x−m)2
N−1
Where: x is each individual score making up the distribution
M is the mean of the distribution
N is the number of scores.
This is illustrated below
Example 1.8 calculation of variance
Calculation of a variance
x x2
x-m, (x-m)2
3 9 -2 4
5 25 0 0
2 4 -3 9
7 49 2 4
9 81 4 16
4 16 -1 1
∑ 30 184 34
M = 30/6 = 5.0
S2 =
34/5 = 6.8
(Keronanton, 2004)
Standard Deviation
Though the variance is frequently used as a measure of spread in certain statistical calculations, it does have
the disadvantage of being expressed in units different from those of those of the summarized data. Which means the
expressed units is going to be far smaller than the data. However the variance can be easily converted into a measure
of spread expressed in the same unit of measurement as the original scores: the standard deviation(s). It should be
noted that Standard deviation indicate the fluctuation of the variables around their mean. To convert from variance to
the standard deviation simply find the square root of the variance. It is the most popular measure of spread. The
formula for the standard deviation is given below.
1)-(N
N
)x(
-x
=S
2
2
5. Example 1.8 calculation of standard deviation
Find the standard deviation of example 1.7
√6.8 = 2.61
Example 1.9
Find the standard deviation of the following distributed values
Times
score
6 3 5 5 2 7 6 4 3 2
Mean
score
4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3
To get the standard deviation, subtract the mean from each of the scores, square the deviation, and then add up the
squared deviations. This process is outlined below.
Time Scores Mean Score - mean (Score - mean)
6 4.3 1.7 2.89
3 4.3 -1.3 1.69
5 4.3 0.7 0.49
5 4.3 0.7 0.49
2 4.3 -2.3 5.29
7 4.3 2.7 7.29
6 4.3 1.7 2.89
4 4.3 -0.3 0.09
3 4.3 -1.3 1.69
2 4.3 -2.3 5.29
Total = 28.10
Therefore, the standard deviation becomes: 1.77=3.12=
9
28.10
=S
Grouped Data:
Often, data will be reported in terms of grouped observation and the calculation of the standard is obtained by a slightly
different formula, which is easier to apply in this situation. An example is presented below:
Example 1.10 Ages f
51 - 60 3
41 - 50 10
31 - 40 15
6. 21 - 30 11
11 - 20 5
To calculate the mean and standard deviation of a grouped data, you must determine the midpoint for each of
the groups of observations. (Panneerselvam, 2008) Adding the upper and lower scores for each interval and dividing
by two can obtain the midpoints. For example, for the first group of data the midpoint would be (51 + 60)/2 = 111/2 =
55.5. I have redrawn the data below with the midpoints inserted. Also, I have included in the redrawn data a column
headed by the term fxMidpoint, which is simply the midpoint multiplied by the frequency
Ages f Midpoints fxMidpoint
51 - 60 3 55.5 166.5
41 - 50 10 45.5 455.0
31 - 40 15 35.5 532.5
21 - 30 11 25.5 280.5
11 - 20 5 15.5 77.5
Total = 1512.0
The mean becomes the sum of the scores in the fxMidpoint column divided by the sample size. The sample size can
be determined by adding the f column (N = 44). Therefore, the mean = 1512.0/44 = 34.36 (rounded to 34.4).To
determine the standard deviation there is a need to add one additional column to the table of calculations above. This
additional column is fxmidpoint2
and I have redrawn our table below with the added column.
Ages f Midpoints fxMidpoint fxMidpoint2
51 - 60 3 55.5 166.5 9240.75
41 - 50 10 45.5 455.0 20702.50
31 - 40 15 35.5 532.5 18903.75
21 - 30 11 25.5 280.5 7152.75
11.04=121.93=
43
51957.82-57201
=
43
44
01512.
-57201.0
=S
2
7. Chapter Exercises
1. A magazine is interested in expanding its readership to "yuppies," defined as people between the ages of
30 to 40. Following are the ages of a random selection of the magazine's readership, is there any reason to be
concerned? Draw the theoretical distribution and contrast the actual score distribution with the theoretical
distribution. Is the sample adequate?
23, 31, 29, 21, 25, 27, 25, 21, 29, 30, 35, 41, 23, 35,
19, 20, 26, 24, 26, 25, 28, 27, 51, 15, 28, 21, 23, 25
2. A more extensive examination of the readership was undertaken after the initiation of a one-year advertising
program, designed to increase the readership age range. Following are the data collected from this more
extensive study. Draw the theoretical distribution of the ages of the readers. What do you conclude? Is the
sample adequate?
Ages Frequency
20 - 24 26
25 - 29 45
30 - 34 87
35 - 39 30
40 - 44 8
45 - 49 4
References
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Panneerselvam, R. (2008). Research Method. New delhi: Prentice hall of india limited.
Schuetter, J. (2007). Chapter 1. In J. Schuetter, measures of dispersation (pp. 45-54).
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