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Friday, February 14, 2020

Continuous Variable and Categorical Variable| Introduction to Educational Statistics | BEd Solved Assignment Course Code 8614


 Explain the following terms with examples.  

a)  Continuous Variable                     

b)  Categorical Variable

c)  Independent Variable

d)  Dependent Variable

e)  Co-Variation


  • Course: Introduction to Educational Statistics (8614)
  • Level: B.Ed (1.5 Years)

Answer:

A variable is a quantity that has a changing value; the value can vary from one example to the next. A continuous variable is a variable that has an infinite number of possible values. In other words, any value is possible for the variable.  A continuous variable is the opposite of a discrete variable, which can only take on a certain number of values.  A continuous variable doesn’t have to have every possible number (like  -infinity to +infinity), it can also be continuous between two numbers, like 1 and 2. For example, discrete variables could be 1,2 while the continuous variables could be 1,2 and everything in between: 1.00, 1.01, 1.001, 1.0001…


What is a Continuous Variable? Examples of Continuous Data



A few examples of continuous variables/data:

•  Time it takes a computer to complete a task. You might think you can count it, but time is often rounded up to convenient intervals, like seconds or milliseconds. Time is actually a continuum: it could take 1.3 seconds or it could take 1.333333333333333… seconds.


•  A person’s weight. Someone could weigh 180 pounds, they could weigh 180.10 pounds or they could weigh 180.1110 pounds. The number of possibilities for weight is limitless.

_______________________________________________________________________________

b)  Categorical Variable

Answer:


In statistics, a categorical variable is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category based on some qualitative property.[1] In computer science and some branches of mathematics, categorical variables are referred to as enumerations or enumerated types. Commonly (though not in this article), each of the possible values of a categorical variable is referred to as a level. The probability distribution associated with a random categorical variable is called a categorical distribution.


Categorical data is the statistical data type consisting of categorical variables or data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cross-tabulations, or from observations of quantitative data grouped within given intervals. Often, purely categorical data are summarised in the form of a contingency table.


However, particularly when considering data analysis, it is common to use the term "categorical data" to apply to data sets that, while containing some categorical variables, may also contain non-categorical variables.

______________________________________________________________________________

c)  Independent Variable


Answer:


INDEPENDENT VARIABLE DEFINITION


An independent variable is defined as a variable that is changed or controlled in a scientific experiment. It represents the cause or reason for an outcome.  Independent variables are the variables that the experimenter changes to test their dependent variable.  A change in the independent variable directly causes a change in the dependent variable. The effect on the dependent variable is measured and recorded.

Common Misspellings: independent variable

INDEPENDENT VARIABLE EXAMPLES

•  A scientist is testing the effect of light and dark on the behavior of moths by turning a light on and off. The independent variable is the amount of light and the moth's reaction is the dependent variable.
•  In a study to determine the effect of temperature on plant pigmentation, the independent variable (cause) is the temperature, while the amount of pigment or color is the dependent variable (the effect).

GRAPHING THE INDEPENDENT VARIABLE

When graphing data for an experiment, the independent variable is plotted on the x-axis, while the dependent variable is recorded on the y-axis. An easy way to keep the two variables straight is to use the acronym DRY MIX, which stands for:
• The dependent variable that Responds to change goes on the Y-axis
•  Manipulated or Independent variable goes on the X-axis
____________________________________________________________________________ 

d)  Dependent Variable


Answer:


The two main variables in an experiment are the independent and dependent variables. An independent variable is a variable that is changed or controlled in a scientific experiment to test the effects on the dependent variable.  A dependent variable is the variable being tested and measured in a scientific experiment.  The dependent variable is 'dependent' on the independent variable. As the experimenter changes the independent variable, the effect on the dependent variable is observed and recorded.

For example, a scientist wants to see if the brightness of light has any effect on a moth being attracted to the light. The brightness of the light is controlled by the scientist. This would be the independent variable. How the moth reacts to the different light levels (distance to the light source) would be the dependent variable.

The independent and dependent variables may be viewed in terms of cause and effect. If the independent variable is changed, then an effect is seen in the dependent variable. Remember, the values of both variables may change in an experiment and are recorded. The difference is that the value of the independent variable is controlled by the experimenter, while the value of the dependent variable only changes in response to the independent variable.

When results are plotted in graphs, the convention is to use the independent variable as the x-axis and the dependent variable as the y-axis.
_______________________________________________________________________________ 

e)  Co-Variation


Answer:


When explaining other people’s behaviors, we look for similarities (covariation) across a range of situations to help us narrow down specific attributions. There are three particular types of information we look for to help us decide, each of which can be high or low:
•  Consensus: how similarly other people act, given the same stimulus, as the person in question.
•  Distinctiveness: how similarly the person acts in different situations, towards other stimuli.
•  Consistency: how often the same stimulus and response in the same situation are perceived.

People tend to make internal attributions when consensus and distinctiveness are low but consistency is high. They will make external attributions when consensus and distinctiveness are both high and consistency is still high. When consistency is low, they will make situational attributions.


People are often less sensitive to consensus information.


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Thursday, February 13, 2020

Central Tendency and Measures of Dispersion | Introduction to Educational Statistics | BEd Solved Assignment Course Code 8614

Q.3: Explain the measures of central tendency and measures of dispersion. How these two concepts are related?  How these two concepts are related? Suggest one measure of dispersion for each measure of central tendency with logical reasons.

  • Course: Introduction to Educational Statistics (8614)
  • Level: B.Ed (1.5 Years)

Answer:

Collecting data can be easy and fun. But sometimes it can be hard to tell other people about what you have found. That’s why we use statistics. Two kinds of statistics are frequently used to describe data. They are measures of central tendency and dispersion. These are often called descriptive statistics because they can help you describe your data.

Mean, median, and mode


These are all measures of central tendency. They help summarize a bunch of scores with a single number. Suppose you want to describe a bunch of data that you collected to a friend for a particular variable like the height of students in your class. One way would be to read each height you recorded to your friend. Your friend would listen to all of the heights and then come to a conclusion about how tall students generally are in your class, but this would take too much time. Especially if you are in a class of 200 or 300 students! Another way to communicate with your friend would be to use measures of central tendency like the mean, median, and mode. They help you summarize bunches of numbers with one or just a few numbers. They make telling people about your data easy.

Range, variance, and standard deviation


These are all measures of dispersion. These help you to know the spread of scores within a bunch of scores. Are the scores really close together or are they really far apart? For example, if you were describing the heights of students in your class to a friend, they might want to know how much the heights vary. Are all the men about 5 feet 11 inches within a few centimeters or so? Or is there a lot of variation where some men are 5 feet and others are 6 foot 5 inches? Measures of dispersion like the range, variance, and standard deviation tell you about the spread of scores in a data set. Like central tendency, they help you summarize a bunch of numbers with one or just a few numbers.

How these two concepts are related? Suggest one measure of dispersion for each measure of central tendency with logical reasons.


In many ways, measures of central tendency are less useful in statistical analysis than measures of dispersion of values around the central tendency.  The dispersion of values within variables is especially important in social and political research because:
•       Dispersion or "variation" in observations is what we seek to explain.
•   Researchers want to know WHY some cases lie above average and others below average for a given variable:
•       TURNOUT in voting: why do some states show higher rates than others?
•       CRIMES in cities: why are there differences in crime rates?
•       CIVIL STRIFE among countries: what accounts for differing amounts?
•       Much of statistical explanation aims at explaining DIFFERENCES in observations --  also known as
§  VARIATION, or the more technical term, VARIANCE.
The SPSS Guide contains only the briefest discussion of measures of dispersion on pages 23-24.
•       It mentions the minimum and maximum values as the extremes, and
•       it refers to the standard deviation as the "most commonly used" measure of dispersion.
This is not enough, and we'll discuss several statistics used to measure variation, which differ in their importance.
•       We'll proceed from the less important to the more important, and
•       we'll relate the various measures to measurement theory.

Easy-to-Understand Measures of dispersion for NOMINAL and ORDINAL variables

In the great scheme of things, measuring dispersion among nominal or ordinal variables is not very important.
•   There is inconsistency in methods to measure dispersion for these variables, especially for nominal variables.
•   Measures suitable for nominal variables (discrete, non-order able) would also apply to discrete order able or continuous variables, order able, but better alternatives are available.
•  Whenever possible, researchers try to re-conceptualize nominal and ordinal variables and operationalize (measure) them with an interval scale.

Variation Ratio, VR

•       VR = l - (proportion of cases in the mode)
•       The value of VR reflects the following logic:
§  The larger the proportion of cases in the mode of a nominal variable, the less the variation among the cases of that variable.
§  By subtracting the proportion of cases from 1, VR reports the dispersion among cases.
o   This measure has an absolute lower value of 0, indicating NO variation in the data (occurs when all the cases fall into one category; hence no variation).
o   Its maximum value approaches one as the proportion of cases inside the mode decreases.
•       Unfortunately, this measure is a "terminal statistic":
§  VR does not figure prominently in any subsequent procedures for statistical analysis.
§  Nevertheless, you should learn it, for it illustrates
o   That even nominal variables can demonstrate variation
o   That the variation can be measured, even if somewhat awkwardly.

Easy-to-understand measures of variation for CONTINUOUS variables.


RANGE: the distance between the highest and lowest values in a distribution
•       Uses information on only the extreme values.
•       Highly unstable as a result.
SEMI-INTERQUARTILE RANGE: distance between scores at the 25th and the 75th percentiles.
•       Also uses information on only two values, but not ones at the extremes.
•       More stable than the range but of limited utility.

AVERAGE DEVIATION:

where |X, -X’|  =  absolute value of the differences
Absolute values of the differences are summed, rather than the differences themselves, for summing the positive and negative values of differences in a distribution calculating from its mean always yields 0.
•       The average deviation is simple to calculate and easily understood.
•   But it is of limited value in statistics, for it does not figure in subsequent statistical analysis.

•  For mathematical reasons, statistical procedures are based on measures of dispersion that use SQUARED deviations from the mean rather than absolute deviations.

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Wednesday, February 12, 2020

Course Code 8614 | What is normal distribution? Explain the role of normal distribution in decision making for data analysis. Write a note on Skeweness and Kurtosis and explain its causes. | Introduction to Educational Statistics | BEd Solved Assignment Course Code 8614

What is normal distribution? Explain the role of normal distribution in decision-making for data analysis. Write a note on Skewness and Kurtosis and explain its causes.

Course: Introduction to Educational Statistics (8614) 

Level: B.Ed (1.5 Years)


Answer:


In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have nearly normal distributions. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.  The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

The normal distribution is widely observed. Furthermore, it frequently can be applied to situations in which the data is distributed very differently. This extended applicability is possible because of the central limit theorem, which states that regardless of the distribution of the population, the distribution of the means of random samples approaches a normal distribution for a large sample size.

Applications to Business Administration


The normal distribution has applications in many areas of business administration. For example:
·      Modern portfolio theory commonly assumes that the returns of a diversified asset portfolio follow a  normal distribution.


·          In operations management, process variations often are normally distributed.


·      In human resource management, employee performance sometimes is considered to be normally distributed.


The normal distribution often is used to describe random variables, especially those having symmetrical, unimodal distributions. In many cases, however, the normal distribution is only a rough approximation of the actual distribution. For example, the physical length of a component cannot be negative, but the normal distribution extends indefinitely in both the positive and negative directions. Nonetheless, the resulting errors may be negligible or within acceptable limits, allowing one to solve problems with sufficient accuracy by assuming a normal distribution.

Skewness and Kurtosis

Skewness and kurtosis are two commonly listed values when you run a software’s descriptive statistics function.   Many books say that these two statistics give you insights into the shape of the distribution.

Skewness is a measure of the symmetry in a distribution. An asymmetrical dataset will have a skewness equal to 0.   So, a normal distribution will have a skewness of 0. Skewness essentially measures the relative size of the two tails.

Kurtosis is a measure of the combined sizes of the two tails. It measures the amount of probability in the tails.  The value is often compared to the kurtosis of the normal distribution, which is equal to 3.   If the kurtosis is greater than 3, then the dataset has heavier tails than a normal distribution (more in the tails).   If the kurtosis is less than 3, then the dataset has lighter tails than  a normal distribution (less in the tails).   Careful here.   Kurtosis is sometimes reported as “excess kurtosis.”   Excess kurtosis is determined by subtracting 3 from the kurtosis.  This makes the normal distribution kurtosis equal 0.  Kurtosis originally was thought to measure the peakedness of a distribution. Though you will still see this as part of the definition in many places, this is a misconception.

Skewness and kurtosis involve the tails of the distribution.  These are presented in more detail below.

SKEWNESS

Skewness is usually described as a measure of a dataset’s symmetry – or lack of symmetry.   A perfectly symmetrical data set will have a skewness of 0.    The normal distribution has a skewness of 0.
The skewness is defined as (Advanced Topics  in Statistical Process Control, Dr. Donald Wheeler,


where n is the sample size, Xi is the ith value, X is the average and s is the sample standard deviation.   Note the exponent in the summation.   It is “3”.   The skewness is referred to as the “third standardized central moment for the probability model.”

Most software packages use a formula for the skewness that takes into account sample size:




This sample size formula is used here.   It is also what Microsoft Excel uses.   The difference between the two formula results becomes very small as the sample size increases.

Figure 1 is a symmetrical data set.   It was created by generating a set of data from 65 to 135 in steps of 5 with the number of each value as shown in Figure 1.    For example, there are 3 65’s, 6 65’s, etc.

Related Topics

Chi-Square, and independent test.

ANOVA and its Logics

Median (Procedure of Determination, Merits, Demerits)

Measures of Dispersion

Descriptive and Inferential Statistics

What is data Cleaning? Importance and Benefits of Data Cleaning 

Explain the terms Degree of Freedom,Spread of Score,Sample,Z Score,Confidence Interval 

What is measure of difference? Explain different types of test

Concept of Reliability, Types and methods of Reliability

Level of Measurement

Types of Variable in Stats 

Measures of Central Tedency and Dispersion, 

Role of Normal Distribution, and also note on Skewness and Kurtosis. 

Methods of Effective Presentation



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Explain different methods of effective presentation of data. List different types of graphs and write note on each type| Introduction to Educational Statistics | BEd Solved Assignment Course Code 8614


Explain different methods of effective presentation of data. List different types of graphs and write a note on each type.

  • Course: Introduction to Educational Statistics (8614)
  • Level: B.Ed (1.5 Years)

Answer:


The presentation of data is tricky. Not everyone in your audience likes to crunch numbers. Learn 5 ways to make your audience understand your message in 2 seconds or less.

Numbers are distracting

When you present numbers on your slides, you can expect two types of reactions from your audience. One set of audience hates numbers and tunes off. Another set loves to crunch numbers and take off on a tangent. As a presenter, you lose either way unless you know how to guide your audience's attention by making your message obvious. You can also present information creatively to make it interesting.

Here are the 5 tips to present your key message in 2 seconds.

1. Use simple 2D charts instead of complex 3D charts

We don’t doubt the fact that 3D charts look cool. But, when you use  3D, you make your audience work hard. You give them an additional dimension to think about. This delays their understanding.

Let’s do a quick makeover of a 3D chart to convey the key message in under 2 seconds:





The slide looks very colorful but complex. The chart says which product performed how well in each month over the past 6 months. Phew! That’s a lot to grasp at one time. Avoid any Presentation tips that require you to make information complex.

Consider this alternative presentation of the same data:





We used a simple 2D line graph to show the trend over time. The title gives a clear idea of what to look for in the slide. In 2 seconds your audience ‘gets’ the message of the slide.
To learn 29 creative ways to present data and other components of your presentations creatively, check the free Creative Presentation Ideas e-course.

2. Use labels instead of legends

Take a look at this slide with a pie chart:





Though it’s a simple chart to grasp, the legends placed off the chart delay understanding. Your audience needs to refer to the legends each time to make sense of the colors.

Consider this alternative pie chart:





The audience can find all the relevant information in one place instead of having to search around
the slide. The slide title gives the core message. The relevant part of the pie chart is isolated
for easy reference. So, your audience ‘gets’ your message under 2 seconds.

3. Make your key point stand out

Take a look at this slide with data:




Can you tell me what the key message of the slide is? Neither can your audience. Slides without a clear focus take a long time to understand. You can read further tips for data presentation here.
Consider this alternative slide with a graph:





The key point almost jumps out of the slide. To make a presentation of data effective, we ruthlessly removed everything that could potentially distract the audience's attention. There are no grid lines. Units on the y-axis are replaced by data labels. The key number is made larger than the rest. Naturally, your audience gets the message in under 2 seconds.

Different types of graphs:

There are different types of graphs in mathematics and statistics that are used to represent data in a pictorial form. Among the various types of charts or graphs, the most common and the most widely used ones are given and explained below.

Types of Graphs and Charts

•        Statistical Graphs (bar graph, pie graph, line graph, etc.)
•        Exponential Graphs
•        Logarithmic Graphs
•        Trigonometric Graphs
•        Frequency Distribution Graph
All these graphs are used in various places to represent a certain set of data concisely. The details of each of these graphs (or charts) are explained below in detail which will not only help to know about these graphs better but will also help to choose the right kind of graph for a particular data set.

Statistical Graphs

A statistical graph or chart is defined as the pictorial representation of statistical data in graphical form. Statistical graphs are used to represent a set of data to make it easier to understand and interpret statistical data.

Exponential Graphs

Exponential graphs are the representation of exponential functions using the table of values and plotting the points on graph paper. It should be noted that the exponential functions are the inverse of logarithmic functions. In the case of exponential charts, the graph can be an increasing or decreasing one based on the function. An example is given below which will help to understand the concept of graphing exponential function easily.
For example, the graph of y = 3x is an increasing one while the graph of y = 3-x is a decreasing one.




Logarithmic Graphs

Logarithmic functions are inverse of exponential functions and the methods of plotting them are similar. To plot logarithmic graphs, it is required to make a table of values and then plot the points accordingly on graph paper. The graph of any log function will be the inverse of an exponential function. An example is given below for a better understanding.

For example, the inverse graph of y = 3x will be y = log3 {x) which will be as follows:




Trigonometric Graphs

Trigonometry graphs are plotted for the 6 trigonometric functions which include sine function, cosine function, tangent function, cotangent function, cosec function, and sec function. Visit trigonometry graphs to learn the graphs of each of the functions in detail along with their maximum and minimum values and solved examples.



Frequency Distribution Graph

A frequency distribution graph is used to show the frequency of the outcomes in a particular sample. For frequency distribution graphs, the table of values is made by placing the outcomes in one column and the number of times they appear (i.e. frequency) in the other column.

This table is known as the frequency distribution table. There are two commonly used frequency graphs which include: 

  • Frequency Polygon        
  • Cumulative Frequency Distribution Graphs



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