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Chapter 1: Statistics, Computers and Statistical Packages

Basic Definitions

Samples and Populations

Samples refers to some subset of a population and a population is some large set of

numbers

o There are 3 types of populations

Finite population: has a definite number of individuals

Infinite population: has no limit on the number of individuals

Theoretical population: is simply an equation

The difference between a random sample and a non-random sample

o Random sample is one in which every observation in the population has an equal

opportunity of being selected

Inferential statistics are all based on the assumption of random sampling

o A non-random sample is one in which this is not the case

Statistics and Parameters

Statistic: is a number that describes a sample

o Ex. the mean, standard deviation and the variance

Parameter: is a number that describes a population

o µ is the mean, σ is the standard deviation and σ2 is the variance

Difference between the parameter and the statistic

o We are usually unable to assess the population

o Instead we have to make do with a sample from the population and use the

information we obtain on the sample to estimate the corresponding value of the

population

We have a statistic and wish to estimate the parameter

Unbiased and Biased Estimates

The difference between biased and unbiased estimates of parameters

A statistic is said to be unbiased if the mean of all possible values of that statistic is equal

to the parameter

o The mean is a statistic that is unbiased

o This is because the mean of the means of all possible samples from the population

equals the population mean

A statistic is said to be biased if the mean of all possible values is not equal to the

population value

o The variance is said to be biased because the mean of the variance of all possible

samples from a population is less than the population variance

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Types of Statistics

Statistics of Location

Statistics serve to locate the sample on the number line, which is a representation of all

possible numerical values ranging from minus infinity to plus infinity

Any statistic that helps you locate where the sample is on this number line is a statistic of

location

One class of such statistics are the 3 measures of central tendency

o Median

The median is that value such that 50% of the values are greater than that

value and 50% are less

It is the value that is at the centre of all the values

o Mode

The mode is that value that occurs most frequently and the most popular

one

If 2 values had the same highest frequency, the distribution of the scores

are called bimodal and if more it is called multi-modal

o Mean

The mean is that value such that the sum of the deviations of the scores

from their mean and add to 0

It is identified as

The formula is

Statistics of Scale

Statistics of scale describe how much differentiation there is in a sample

o Sometimes called statistics of variation or dispersion

If all the numbers are close together, the scale is small and if big, the scale is large

They give an indication of the amount of variability in a sample

There are a few statistics of scale

o Range

The simplest measure of scale is range

It is defined as the difference between the highest and the lowest value

However that makes it a problem measure since it only takes 2 values into

account

Measures that use all of the numbers in the sample would be expected to

give much more stable answers

o Semi-interquartile range

A type of range statistic that uses more information in the distribution is

the semi-interquartile range, also known as the quartile deviation

Is it defined as the difference between the 75th and the 25th percentile

divided by 2

This makes it more stable than the range

o Absolute deviations

Mean absolute deviation

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Mean absolute deviation is computed by summing the absolute

deviations and dividing by the mean of deviations

This mean would give an indication of the relative size of the

deviations of the values from the mean

Median absolute deviations

The median absolute deviation is simply the median of the

absolute deviations

This tells us that roughly 50% of the values differ more than this

much from the mean, while 50% of the values deviate less than

this

o Variance and standard deviation

Variance is the squared deviations of the values from the mean and then

calculate the mean of these squared deviations

It is generally identified as S2

The formula is biased formula is

o If you wished to describe the variance or the standard

deviation of the sample, you would use the biased estimate

The formula for the unbiased estimate of the population variance is

defined as

o If you wished to used your statistic to estimate the

population variance or the standard deviation of the sample,

you would use the unbiased estimate

The square root of the variance is referred to as the standard deviation

and it is identified as S

Statistic of Shape

A statistic of shape tells us how the values are distributed along the line, whether they are

symmetrically distributed around the mean or skewed to one end of the other

A standard score is identified by the letter Z and it is defined as

o S is the biased estimate of the variance and Z values are a transformation of the

original X values, such that the mean of the Z is 0 and the variance is 1

The statistic of shape are:

o Skewness

Skewness is a measure of asymmetry of the distribution of numbers

It is identified as g1 and the formula is

Cubing a large deviation yields a large number and retains the sign of the

deviation, thus if g1 is

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