AIOU Assignment BEd 1.5 Year 2.5 Year 8614 Educational Statistics Assignment 1

AIOU Assignment BEd 1.5 Year 2.5 Year 8614 Educational Statistics Assignment 1

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Q.1 Explain different methods of effective presentation of data. List different types of graphs and write note on each type.
Answer:

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 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 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 requires you to use 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 delays understanding. Your audience needs to refer to the legends each time to make sense of the colors.

Consider this alternative pie chart: 

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 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 graph: 

The key point almost jumps out of the slide. To make presentation of data effective, we ruthlessly removed everything that can potentially distract the audience 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 under 2 seconds.

Different types of graphs:

There are different types of graphs in mathematics and statistics which 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 in a concise way. 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. The 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 a 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 functionin an easy way.

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 method 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 a graph paper. The graph of any log function will be the inverse of an exponential function. An example is given below for 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 function 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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Q.2 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.
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 distributions that are nearly normal. 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 a widely observed distribution. 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.

Skeweness 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. A symmetrical 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 form 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, www.spcpress.com):

where n is the sample size, Xi is the ith X 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..

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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
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 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:

1. Dispersion or "variation" in observations is what we seek to explain.

2. 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?

3. 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 norminal or oridinal 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-orderable) would also apply to discrete orderable or continuous variables, orderable, but better alternatives are available.
• Whenever possible, researchers try to reconceptualize nominal and ordinal variables and operationalize (measure) them with an interval scale.

Variation Ratio, VR

1.  VR = l - (proportion of cases in the mode)

2.  The value of VR reflects the following logic:

a. The larger the proportion of cases in the mode of a nominal variable, the less the variation among the cases of that variable.

b. By subtracting the proportion of cases from 1, VR reports the dispersion among cases.

• 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).
• Its maximum value approaches one as the proportion of cases inside the mode decreases.

3. Unfortunately, this measure is a "terminal statistic":

a VR does not figure prominently in any subsequent procedures for statistical analysis.

b Nevertheless, you should learn it, for it illustrates

• that even nominal variables can demonstrate variation
• 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

= 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.
• or 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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Q.4 Explain the following terms with examples. (20)
a) Continuous Variable
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 are limitless.

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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 on the basis of 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 of 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.

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c) Independent Variable
Answer:

INDEPENDENT VARIABLE DEFINITION

An independent variable is defines as the 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: independant 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:

• Dependent variable that Responds to change goes on the Y axis
• Manipulated or Independent variable goes on the X axis

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d) Dependent Variable
Answer:

The two main variables in an experiment are the independent and dependent variable.

An independent variable is the 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 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.

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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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Q.5 What are levels of measurement? Explain each level so that reader can understand the description of level and differentiate each level from other levels. Write down 10 examples for each level and further explain one example from level
Answer:

Data Levels of Measurement

A variable has one of four different levels of measurement: Nominal, Ordinal, Interval, or Ratio. (Interval and Ratio levels of measurement are sometimes called Continuous or Scale). It is important for the researcher to understand the different levels of measurement, as these levels of measurement, together with how the research question is phrased, dictate what statistical analysis is appropriate. In fact, the Free download below conveniently ties a variable’s levels to different statistical analyses.

In descending order of precision, the four different levels of measurement are:

• Nominal–Latin for name only (Republican, Democrat, Green, Libertarian)
• Ordinal–Think ordered levels or ranks (small–8oz, medium–12oz, large–32oz)
• Interval–Equal intervals among levels (1 dollar to 2 dollars is the same interval as 88 dollars to 89 dollars)
• Ratio–Let the “o” in ratio remind you of a zero in the scale (Day 0, day 1, day 2, day 3, …) 

The first level of measurement is nominal level of measurement. In this level of measurement, the numbers in the variable are used only to classify the data. In this level of measurement, words, letters, and alpha-numeric symbols can be used. Suppose there are data about people belonging to three different gender categories. In this case, the person belonging to the female gender could be classified as F, the person belonging to the male gender could be classified as M, and transgendered classified as T. This type of assigning classification is nominal level of measurement.

The second level of measurement is the ordinal level of measurement. This level of measurement depicts some ordered relationship among the variable’s observations. Suppose a student scores the highest grade of 100 in the class. In this case, he would be assigned the first rank. Then, another classmate scores the second highest grade of an 92; she would be assigned the second rank. A third student scores a 81 and he would be assigned the third rank, and so on. The ordinal level of measurement indicates an ordering of the measurements.

The third level of measurement is the interval level of measurement. The interval level of measurement not only classifies and orders the measurements, but it also specifies that the distances between each interval on the scale are equivalent along the scale from low interval to high interval. For example, an interval level of measurement could be the measurement of anxiety in a student between the score of 10 and 11, this interval is the same as that of a student who scores between 40 and 41. A popular example of this level of measurement is temperature in centigrade, where, for example, the distance between 94 C and 96 C is the same as the distance between 100 C and 102 C.

The fourth level of measurement is the ratio level of measurement. In this level of measurement, the observations, in addition to having equal intervals, can have a value of zero as well. The zero in the scale makes this type of measurement unlike the other types of measurement, although the properties are similar to that of the interval level of measurement. In the ratio level of measurement, the divisions between the points on the scale have an equivalent distance between them.

The researcher should note that among these levels of measurement, the nominal level is simply used to classify data, whereas the levels of measurement described by the interval level and the ratio level are much more exact.

What level of measurement is used for psychological variables?

Rating scales are used frequently in psychological research. For example, experimental subjects may be asked to rate their level of pain, how much they like a consumer product, their attitudes about capital punishment, their confidence in an answer to a test question. Typically these ratings are made on a 5-point or a 7-point scale. These scales are ordinal scales since there is no assurance that a given difference represents the same thing across the range of the scale. For example, there is no way to be sure that a treatment that reduces pain from a rated pain level of 3 to a rated pain level of 2 represents the same level of relief as a treatment that reduces pain from a rated pain level of 7 to a rated pain level of 6.

In memory experiments, the dependent variable is often the number of items correctly recalled. What scale of measurement is this? You could reasonably argue that it is a ratio scale. First, there is a true zero point: some subjects may get no items correct at all. Moreover, a difference of one represents a difference of one item recalled across the entire scale. It is certainly valid to say that someone who recalled 12 items recalled twice as many items as someone who recalled only 6 items.

But number-of-items recalled is a more complicated case than it appears at first. Consider the following example in which subjects are asked to remember as many items as possible from a list of 10. Assume that (a) there are 5 easy items and 5 difficult items, (b) half of the subjects are able to recall all the easy items and different numbers of difficult items, while (c) the other half of the subjects are unable to recall any of the difficult items but they do remember different numbers of easy items. Some sample data are shown below.

Let's compare (1) the difference between Subject A's score of 2 and Subject B's score of 3 with (2) the difference between Subject C's score of 7 and Subject D's score of 8. The former difference is a difference of one easy item; the latter difference is a difference of one difficult item. Do these two differences necessarily signify the same difference in memory? We are inclined to respond "No" to this question since only a little more memory may be needed to retain the additional easy item whereas a lot more memory may be needed to retain the additional hard item. The general point is that it is often inappropriate to consider psychological measurement scales as either interval or ratio.

Consequences of level of measurement

Why are we so interested in the type of scale that measures a dependent variable? The crux of the matter is the relationship between the variable's level of measurement and the statistics that can be meaningfully computed with that variable. For example, consider a hypothetical study in which 5 children are asked to choose their favorite color from blue, red, yellow, green, and purple. The researcher codes the results as follows:

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