Qualitative variable

Qualitative variable

·         This data describes the quality of something in a non-numerical format.

·         Counts can be applied to qualitative data, but you cannot order or measure this type of variable. Examples are gender, marital status, geographical region of an organization, job title….

·         Qualitative data is usually treated as Categorical variable.

With categorical data, the observations can be sorted according into non-overlapping categories or by characteristics. For example, shirts can be sorted according to color; the characteristic 'color' can have non-overlapping categories: white, black, red, etc.  People can be sorted by gender with categories male and female. Categories should be chosen carefully since a bad choice can prejudice the outcome. Every value of a data set should belong to one and only one category.

·         Analyze qualitative data  using:

1      Frequency tables

2      Modes - most frequently occurring

         3      Graphs:  Bar Charts and Pie Charts

What is the importance of sampling?

The importance of sampling is that we can determine the adequate respondents from the total number of target population. Thus, it will be used in the research study which should be adequate to warrant generalization of the findings to the target population. And the sample size represents the characteristics of the whole population. The advantages of sampling are:
1.It is economical and practical; faster and cheaper;
2. It can yield more comprehensive information;
3. It is more accurate; and because of savings
4.It permits in time and money,
5. the sample survey makes possible the use of much larger and much more varied populations than would be possible for the same expenditure if one were making a complete enumeration.
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Moments generating function

Definition

Let be a random variable. If the expected value exists and is finite for all real numbers belonging to a closed interval , with , then we say that possesses a moment generating function and the functionis called the moment generating function of

Definition

Let X be a random variable for an experiment taking values in a subset S of R. The moment generating function of X is the function M_X defined by

M_X(t) = E[exp(tX)] for t in R

Properties

In the following exercises, assume that the moment generating functions are finite in an interval about 0.3. Show that for any nonnegative integer n,

M_X⁽ⁿ⁾(0) = E(Xⁿ)

Thus, the derivatives of the moment generating function at 0 determine the moments of the variable (hence the name).
4. If a and b are constants, show that

M_{aX + b}(t) = exp(bt) M_X(at)

 5. Suppose that X and Y are independent. Show that

M_{X + Y}(t) = M_X(t) M_Y(t)

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Definition of Dispersion

According to Bowley
"Dispersion is the measure of the variation of the items."

According to R L Connar

"Dispersion is the mean of extent to which the individual items vary."

Measures of central tendency

Definition:
A measure of central tendency is a typical value around which other figures congregate that attempts to describe a set of data by identifying the central position within that set of data.
As such, measures of central tendency are sometimes called measures of central location.

Types of measures of central tendency:
The most commonly used measure of central tendency are
1. Mean
2. Median
3. Mode

1.Mean: Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average.
If we have n values in a data set and they have values x1, x2, ..., xn, then the sample mean, usually denoted by (pronounced x bar), is:

2.Median: Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of the two middle numbers.
3.Mode: Mode is the value that occurs most frequently in a set of data.

Examples of Measures of Central Tendency
For the data 1, 2, 3, 4, 5, 5, 6, 7, 8 the measures of central tendency are
Mean = (1+2+3+4+5+6+7+8)/8=4.56
Median = 5
Mode = 5


The mean can again be of three types
1. Arithmetic mean
2. Geometric mean
3. Harmonic mean

1. Arithmetic mean:
A mathematical representation of the typical value of a series of numbers, computed as the sum of all the numbers in the series divided by the count of all numbers in the series.

Arithmetic mean is commonly referred to as "average" or simply as "mean".

Use of statistics

There are huge use of statistics. Such as (1)Statistics helps in providing a better understanding and exact description of a phenomenon of nature. (2) Statistical helps in proper and efficient planning of a statistical inquiry in any field of study. (3) Statistical helps in collecting an appropriate quantitative data.

(4) Statistics helps in presenting complex data in a suitable tabular, diagrammatic and graphic form for an easy and clear comprehension of the data. (5) Statistics helps in understanding the nature and pattern of variability of a phenomenon through quantitative observations. (6) Statistics helps in drawing valid inference, along with a measure of their reliability about the population parameters from the sample data.

Scope of statistics

Some consider statistics to be a mathematical body of science pertaining to the collection, analysis, interpretation or explanation, and presentation of data,[5] while others consider it a branch of mathematics concerned with collecting and interpreting data. Because of its empirical roots and its focus on applications, statistics is usually considered to be a distinct mathematical science rather than a branch of mathematics. Much of statistics is non-mathematical: ensuring that data collection is undertaken in a way that allows valid conclusions to be drawn; coding and archiving of data so that information is retained and made useful for international comparisons of official statistics; reporting of results and summarized data (tables and graphs) in ways that are comprehensible to those who need to make use of them; implementing procedures that ensure the privacy of census information.

Statisticians improve the quality of data with the design of experiments and survey sampling. Statistics also provides tools for prediction and forecasting using data and statistical models. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences, government, and business. Statistical consultants are available to provide help for organizations and companies without direct access to expertise relevant to their particular problems.

Statistical methods can be used for summarizing or describing a collection of data; this is called descriptive statistics. This is useful in research, when communicating the results of experiments. In addition, patterns in the data may be modeled in a way that accounts for randomness and uncertainty in the observations, and are then used for drawing inferences about the process or population being studied; this is called inferential statistics. Inference is a vital element of scientific advance, since it provides a means for drawing conclusions from data subject to random variation.

To prove the guiding theory further, these predictions are tested as well, as part of the scientific method. If the inference holds true, then the descriptive statistics of the new data increase the soundness of that hypothesis. Descriptive statistics and inferential statistics (a.k.a., predictive statistics) together comprise applied statistics.[9][verification needed]

Statistics is closely related to probability theory, with which it is often grouped; the difference is roughly that in probability theory, one starts from the given parameters of a total population to deduce probabilities pertaining to samples, but statistical inference moves in the opposite direction, inductive inference from samples to the parameters of a larger or total population.

What is Statistics?

Statistics is collection organization, presentation, analysis and incineration of numerical data.
Even though you may not have realized it, you probably have made some statistical statements in your everyday conversation or thinking. Statements like "I sleep for about eight hours per night on average" and "You are more likely to pass the exam if you start preparing earlier" are actually statistical in nature.

 Statistics is a discipline which is concerned with:

1. designing experiments and other data collection,
2. summarizing information to aid understanding,
3. drawing conclusions from data, and
4. estimating the present or predicting the future.

 The two statements at the beginning illustrate some of these points.

 In making predictions, Statistics uses the companion subject of Probability, which models chance mathematically and enables calculations of chance in complicated cases.

 Today, statistics has become an important tool in the work of many academic disciplines such as medicine, psychology, education, sociology, engineering and physics, just to name a few. Statistics is also important in many aspects of society such as business, industry and government. Because of the increasing use of statistics in so many areas of our lives, it has become very desirable to understand and practice statistical thinking. This is important even if you do not use statistical methods directly. 

Types of Central tendency

Central tendency

Types of central tendency:

The most commonly used measure of central tendency are

1. Mean

2. Median

3. Mode

The mean can again be of three types

1. Arithmetic mean

2. Geometric mean

3. Harmonic mean