 # Statistics 1.0

## Help Topics

### Quick Start

You can download the software online from The Statistics 1.0 Website. Unzip the software and run Statistics. The features in the main menu are demonstrated by over 100 examples. For any item in the menu, simply select an appropriate example to see how to input the statistical samples and parameters. The examples have been carefully designed to demonstrate all aspects of the software. For details of specific features, read on.
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### Introduction

Select Introduction - A brief introduction to statistics... from the main menu. Statistics is concerned with methods of designing and evaluating statistical experiments to obtain information about practical problems, for example, inspection of quality of raw material or manufactured products, the comparision of machines and tools used in production, the varying market price of stocks and shares, the yield of crop varieties under different conditions, and so on. For instance, in a production process, the differences in the quality of products is variation due to numerous factors whose influence cannot be predicted, so the variation must be regarded as a random variation. In most cases, the inspection of each item would be too expensive and time-consuming. It may even be impossible if it leads to the destruction of the item. Hence instead of inspecting all the items just a few of them (a sample) are inspected and from this inspection conclusions can be drawn about the totality (the population). All these intuitive notions are made precise using probability theory. A classic example of a statistical experiment called a Galton board, is shown on the left. As the ball drops, the probabilities that it will be deflected to the left or right at each triangular obstacle are equal. The balls are collected in receptacles numbered from left to right. Press the button 100 Trials to generate a sample. Statistics 1.0 provides you with all the tools used by professional statisticians to analyze data. To use the program, select an option from the main menu at the top. For each menu option, there are several examples that show you how to use that particular feature of the program.
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### Descriptive Statistics

Select Descriptive Statistics - New sample... from the main menu. A statistical experiment yeilds a sequence of observations x1, x2, ..., xn called a sample. We assume that each sample value xi is a real number. The aim is to represent the sample in a suitable tabular and graphical form. Measures of central tendency and spread are quantities that often give a very good statistical description of the sample.

Write the Sample in its tabbed box and press the button Descriptive Statistics. The program will calculate the following:

• Frequency Table - sample value, frequency, cumulative frequency, relative frequency, cumulative relative frequency and percentile
• Histogram - class intervals, class frequency, relative class frequency
• Measures of Central Tendency - mean, median and modes
• Measures of Spread - variance, standard deviation and interquartile range
The program will also generate the graphs of the frequency function and histogram of the sample. Move the cursor over the graph window to display the coordinates.
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### Statistical Inference

Select Statistical Inference - New sample... from the main menu. A statistical experiment yeilds a sequence of observations x1, x2, ..., xn called a sample. We assume that each sample value xi is a real number drawn randomly from the whole population. The aim is to find a suitable statistical distribution function that fits the sample data. This is the statistical distribution function of the whole population from which the sample was taken. We also determine confidence intervals for the parameters of the statistical distribution function. Thus, from the sample, we infer the probability law that governs the whole population.

Write the Sample in its tabbed box and press the button Statistical Inference. The program will calculate the following:

• Hypothesis - an appropriate statistical distribution that fits the sample data
• Estimation of Parameters - using Fisher's method of maximum likelihood
• Chi-Square Test - goodness of fit of the statistical distribution and statistical inference, with 99% confidence
• Confidence Intervals - for the parameters of the statistical distribution, with 95% confidence
The program will also generate the graphs of the probability function of the statistical distribution and histogram of the sample. Move the cursor over the graph window to display the coordinates.
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### Quality Control

Select Quality Control - New list of samples... from the main menu. No production process is so perfect that all the products are completely alike. There is always a small variation that is caused by uncontrollable factors and must therefore be regarded as a chance variation. It is important to ensure that the products have certain required properties. Random samples of values for a particular required property are taken during the production run, at regular intervals of time. Each time a sample of the same size is taken. The aim is to be able to tell, given this list of samples, whether the production process is running normally, called Quality Control.

Write the List of Samples, the Sample Size, the control Mean and Standard Deviation in their respective tabbed boxes and press the button Quality Control. The program will calculate the following:

• Control Limits - for the mean and standard deviation
• Control Charts Table - sample values, sample mean, sample standard deviation and process status
The program will also generate the graphs of the control charts for the mean and standard deviation. Move the cursor over the graph window to display the coordinates.
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### Acceptance Sampling

Select Acceptance Sampling - New sampling plan... from the main menu. Acceptance Sampling is applied in mass production when a producer is supplying a consumer lots of N items. The decision to accept or reject a particular lot must be made. This decision is based on the result of inspecting a sample of size n from the lot and determining the number of defective items. The producer and consumer agree on a sampling plan, i.e. an acceptance number c such that if the number of defective items in a single sample from the lot exceeds c, then the entire lot is rejected. The producer and consumer may also wish to define the acceptable quality level AQL and rejectable quality level RQL (both numbers between 0 and 1) such that the producer's risk (the chance of rejecting a good lot) and the consumer's risk (the chance of accepting a bad lot) are mutually agreeable.

Write the Lot Size, the Sample Size, the Acceptance Number, the Acceptable Quality Level AQL and the Rejectable Quality Level RQL in their respective tabbed boxes and press the button Acceptance Sampling. The program will calculate the following:

• Single Sampling Plan Table - fraction defective, operating characteristic (OC) and average outgoing quality (AOQ)
• Producer's Risk
• Consumer's Risk
• Average Outgoing Quality Limit (AOQL)
The program will also generate the graphs of the operating characteristic (OC) curve and the average outgoing quality (AOQ) curve. Move the cursor over the graph window to display the coordinates.
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### Regression and Correlation

Select Regression and Correlation - New sample... from the main menu. Regression and correlation analysis is used for statistical experiments in which two quantities are observed simultaneously. One of the variables, X, can be regarded as an ordinary variable which can be measured without any appreciable error. The other variable, Y, is a random variable, and we are interested in the dependence of Y on X. The experimenter first selects X values x1, x2, ..., xn and then observes Y at those values obtaining a sample (x1, y1), (x2, y2), ..., (xn, yn). We assume that the mean of Y is a linear function of X, called the regression line of Y on X.

Write the sample X values and Y values in their respective tabbed boxes and press the button Regression and Correlation. The program will calculate the following:

• The Covariance
• The Correlation
• The Regression Coefficient - maximum likelihood estimate
• The Regression line of Y on X
• A Confidence Interval for the Regression Coefficient - with 95% confidence
The program will also generate the graphs of the sample points and the regression line. Move the cursor over the graph window to display the coordinates.
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### Time Series and Trends

Select Time Series and Trends - New time series... from the main menu. A time series is an ordered sample of observed values for a statistical experiment, the values being observed at regular intervals of time. The aim is to understand the underlying patterns and forecast future values of the time series. We use the method of exponentially smoothed weighted moving averages ESWMA and trends to make short-term forecasts.

Write the Time Series, ESWMA Group Size and ESWMA Smoothing Constant in their respective tabbed boxes and press the button Time Series and Trends. The program will calculate the following:

• The Exponentially Smoothed Weighted Moving Average (ESWMA)
• Trends based upon the most recent terms of the time series
The program will also generate the graphs of the time series and the ESWMA. Move the cursor over the graph window to display the coordinates.
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### Analysis of Variance

Select Analysis of Variance - New list of samples... from the main menu. The procedure used to test equality of means of several normal populations is called Analysis of Variance ANOVA. The procedure involves splitting a total variance into pieces, analyzing it, and then deciding whether to accept or reject equality of the population means based on the relative magnitude of these pieces.

Write the List of Samples and Sample Size in their respective tabbed boxes and press the button Analysis of Variance. The program will calculate the following:

• Analysis of variance table - among populations and residual, calculation of mean squares
• Test of hypothesis - assuming each population is normal with the same variance, test for the equality of means among all populations
• Maximum likelihood estimators - mean and variance
The program will also generate the graphs of the probability and distribution function of the F-Distribution used in testing the hypothesis. Move the cursor over the graph window to display the coordinates.
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## Probability Distributions

### Normal Distribution

Select Probability Distributions - Normal Distribution - New... from the main menu. Write the parameters mean (real number), standard deviation (positive real number) and a list of X values (real numbers) in their respective tabbed boxes and press the button Normal Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Moment generating function
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Poisson Distribution

Select Probability Distributions - Poisson Distribution - New... from the main menu. Write the parameter mean (positive real number) and a list of X values (non-negative integers) in their respective tabbed boxes and press the button Poisson Distribution. The program will calculate the following:
• Range of random variable X
• Probability function f(x)
• Distribution function F(x)
• Mean and variance
• Factorial moment generating function
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Binomial Distribution

Select Probability Distributions - Binomial Distribution - New... from the main menu. Write the parameters n (positive integer), p (real number between 0 and 1) and a list of X values (non-negative integers) in their respective tabbed boxes and press the button Binomial Distribution. The program will calculate the following:
• Range of random variable X
• Probability function f(x)
• Distribution function F(x)
• Mean and variance
• Factorial moment generating function
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Exponential Distribution

Select Probability Distributions - Exponential Distribution - New... from the main menu. Write the parameter mean (positive real number) and a list of X values (positive real numbers) in their respective tabbed boxes and press the button Exponential Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Moment generating function
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Uniform Distribution

Select Probability Distributions - Uniform Distribution - New... from the main menu. Write the parameters a, b (real numbers with a less than b) and a list of X values (real numbers between a and b) in their respective tabbed boxes and press the button Uniform Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Moment generating function
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Hypergeometric Distribution

Select Probability Distributions - Hypergeometric Distribution - New... from the main menu. Write the parameters N (positive integer), M (non-negative integer less than N), n (non-negative integer less than N) and a list of X values (non-negative integers less than n) in their respective tabbed boxes and press the button Hypergeometric Distribution. The program will calculate the following:
• Range of random variable X
• Probability function f(x)
• Distribution function F(x)
• Mean and variance
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Geometric Distribution

Select Probability Distributions - Geometric Distribution - New... from the main menu. Write the parameter p (real number between 0 and 1) and a list of X values (positive integers) in their respective tabbed boxes and press the button Geometric Distribution. The program will calculate the following:
• Range of random variable X
• Probability function f(x)
• Distribution function F(x)
• Mean and variance
• Factorial moment generating function
• Statistical table - x values, f(x) values and F(x) values
• Random sample
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Beta Distribution

Select Probability Distributions - Beta Distribution - New... from the main menu. Write the parameters a, b (positive real numbers) and a list of X values (real numbers between 0 and 1) in their respective tabbed boxes and press the button Beta Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Statistical table - x values, f(x) values and F(x) values
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Gamma Distribution

Select Probability Distributions - Gamma Distribution - New... from the main menu. Write the parameters n, l (positive real numbers) and a list of X values (positive real numbers) in their respective tabbed boxes and press the button Gamma Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Moment generating function
• Statistical table - x values, f(x) values and F(x) values
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Negative Binomial Distribution

Select Probability Distributions - Negative Binomial Distribution - New... from the main menu. Write the parameters n (positive integer), p (real number between 0 and 1) and a list of X values (integers greater than n) in their respective tabbed boxes and press the button Negative Binomial Distribution. The program will calculate the following:
• Range of random variable X
• Probability function f(x)
• Distribution function F(x)
• Mean and variance
• Factorial moment generating function
• Statistical table - x values, f(x) values and F(x) values
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Chi-Square Distribution

Select Probability Distributions - Chi-Square Distribution - New... from the main menu. Write the parameter degrees of freedom m (positive integer) and a list of X values (positive real numbers) in their respective tabbed boxes and press the button Chi-Square Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Moment generating function
• Statistical table - x values, f(x) values and F(x) values
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Student's t-Distribution

Select Probability Distributions - Student's t-Distribution - New... from the main menu. Write the parameter degrees of freedom m (positive integer) and a list of X values (real numbers) in their respective tabbed boxes and press the button Student's t-Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Statistical table - x values, f(x) values and F(x) values
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Fisher's F-Distribution

Select Probability Distributions - Fisher's F-Distribution - New... from the main menu. Write the parameter degrees of freedom m (positive integer) and a list of X values (real numbers) in their respective tabbed boxes and press the button F-Distribution. The program will calculate the following:
• Range of random variable X
• Probability density function f(x)
• Distribution function F(x)
• Mean and variance
• Statistical table - x values, f(x) values and F(x) values
The program will also generate the graphs of the probability and distribution functions. Move the cursor over the graph window to display the coordinates.
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### Print and Export

Right click on the generated Statistics 1.0 document to select Print and Export to Microsoft Excel options.
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