Proba and Stat Lesson - Part 1

8/12/2019 Proba and Stat Lesson - Part 1

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Statistics = is regarded as the Branch of

Mathematics which involved in:

1. The design of an experiments and other science of

data collection,

2. In manipulating and summarizing data which yield in

wise decision making

3. In coming-up with scientific conclusions from theresults of the data interpretation, and aid in

forecasting or predictions.


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Statistics = is an art which involves systematic

collection, presentation, analysis andinterpretation of data which will results to a

meaningful facts and thereby plays an important

roles in decision making.

Branch of Statistics

1. Descriptive Statistics

2. Inferential Statistics


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Descriptive Statistics = (1) Deals with the

presentation and collection of data. This is the

first part any statistical analysis. (2) are a way ofsummarizing data - letting one number stand for

a group of numbers. We can also use tables and

graphs to summarize data.(3) It deals with

collection of data, its presentation in variousforms, such as tables, graphs and diagrams and

findings averages and other measures which

would describe the data.


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Example: Industrial Statistics, Business

Statistics, Housing and Population Statistics,

Stocks, Trade Statistics.


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Inferential Statistics = (1) it deals with

techniques used for analysis of data, making the

estimates and drawing conclusions from limitedinformation taken on sample basis and testing

the reliability of the estimates.. (2) involves

drawing the right conclusions from the

statistical analysis that has been performedusing descriptive statistics.


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Example: Suppose we want to investigate the

effectiveness of certain medicine to cure an

illness, we take a sample from the population andconduct an experiment to two groups of

respondents and compare the results. This will

provide inferences about the population

proportion. This study belongs to inferentialstatistics.


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Measure of Central Tendency

A measure of central tendency is (1). a statistic which

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. (2). To provide an index to describe a group or

the difference between groups

Mean Not applicable to Nominal Data

Sample Mean(x-bar) Population Mean Mu


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Median Middlemost distribution of data. The point in a

distributin of measures below which 50 percent of thecases lie and that the other 50 percent lie above this point

Odd Number

1

2

xmdn

ith x

1 98

2 90

mdn 3 864 84

5 81

5 13

2mdn rd


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Median Middlemost distribution of data. The point in a

distributin of measures below which 50 percent of thecases lie and that the other 50 percent lie above this point

Even Number2

;

2 2

x xmdn mdn

ith x

1 98

2 90

mdn 3 86

mdn 4 84

5 81

6 77

63

2 2

2 6 24

2 2

86 8485

2

xmdn rd

xmdn th

Mdn


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Mode is the value that occurs more frequently. It is

possible to have more than one mode, if there are two

modes the data is said to be bimodal. It is also possible for

a set of data to not have any mode, this situation occurs if

the number of modes gets to be too large". It is not

really possible to define too large" but one should

exercise good judgment. A reasonable, though very

generous, rule of thumb is that if the number of data

points accounted for in the list of modes is half or moreof the data points, then there is no mode.


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When to use the Measures of Central Tendency

Interval /

Ratio Data

Ordinal /

Nominal Data

Mean

Median

Mode


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Skewness

Skewness We often test whether our data is normally

distributed because this is a common assumption

underlying many statistical tests. Some distributions of

data, such as the bell curve are symmetric. This means

that the right and the left are perfect mirror images of one

another. But not every distribution of data is symmetric.

Sets of data that are not symmetric are said to be

asymmetric. The measure of how asymmetric a

distribution can be is called skewness. As we will see, datacan be skewed either to the right or to the left.

.


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Data that are skewed to the right have a long tail that

extends to the right. An alternate way of talking about a

data set skewed to the right is to say that it is positively

skewed. In this situation the mean and the median are

both greater than the mode. As a general rule, most of the

time for data skewed to the right, the mean will be greaterthan the median. In summary, for a data set skewed to the

right:

.

.


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In fact, in any symmetrical distribution the mean, median

and mode are equal. However, in this situation, the mean is

widely preferred as the best measure of central tendency

because it is the measure that includes all the values in the

data set for its calculation, and any change in any of the

scores will affect the value of the mean. This is not thecase with the median or mode.

.

.


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we find that the mean is being dragged in the direct of the

skew. In these situations, the median is generally

considered to be the best representative of the central

location of the data. The more skewed the distribution,

the greater the difference between the median and mean,

and the greater emphasis should be placed on using themedian as opposed to the mean. A classic example of the

above right-skewed distribution is income (salary), where

higher-earners provide a false representation of the

typical income if expressed as a mean and not a median..

.


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If dealing with a normal distribution, and tests of

normality show that the data is non-normal, it is

customary to use the median instead of the mean.

However, this is more a rule of thumb than a strict

guideline. Sometimes, researchers wish to report the mean

of a skewed distribution if the median and mean are notappreciably different (a subjective assessment), and if it

allows easier comparisons to previous research to be made.

.


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Skewed to the right

Always: mode < mean

Always: mode < medianMost of the time: mode < median < mean

.


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SKEWE TO THE LEFT

Always: mean < mode

Always: median < mode

Most of the time: mean < median < mode


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1. Rule One. If the mean is less than the median, thedata are skewed to the left.

2. Rule Two. If the mean is greater than the median,

the data are skewed to the right.


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Kurtosis

Kurtosis Kurtosis is the measure of the peak of a

distribution, and indicates how high the distribution is

around the mean. The kurtosis of a distributions is in one

of three categories of classification.

1. Mesokurtic

2. Leptokurtic

3. Platykurtic

.


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Kurtosis

1. Mesokurtic = Kurtosis is typically measured with respect to thenormal distribution. A distribution that is peaked in the same way as

any normal distribution, not just the standard normal distribution, is

said to be mesokurtic. The peak of a mesokurtic distribution is

neither high nor low, rather it is considered to be a baseline for the

two other classifications.

2. Leptokurtic = A leptokurtic distribution is one that has kurtosis

greater than a mesokurtic distribution. Leptokurtic distributions are

identified by peaks that are thin and tall. The tails of these

distributions, to both the right and the left, are thick and heavy.

Leptokurtic distributions are named by the prefix "lepto" meaning

"skinny."

.


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Kurtosis

1. Platikurtic = distributions are those that have a peak lower than amesokurtic distribution. Platykurtic distributions are characterized

by a certain flatness to the peak, and have slender tails. The name of

these types of distributions come from the meaning of the prefix

"platy" meaning "broad.

.


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SKEWE TO THE LEFTThe situation reverses itself when we deal with data

skewed to the left. Data that are skewed to the left

have a long tail that extends to the left. An alternate

way of talking about a data set skewed to the left isto say that it is negatively skewed. In this situation

the mean and the median are both less than the

mode. As a general rule, most of the time for data

skewed to the left, the mean will be less than the

median. In summary, for a data set skewed to the

left:

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Proba and Stat Lesson - Part 1