Proba and Stat Lesson - Part 1
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Transcript of 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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Branch of Statistics
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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Branch of Statistics
Example: Industrial Statistics, Business
Statistics, Housing and Population Statistics,
Stocks, Trade Statistics.
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Branch of Statistics
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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Branch of Statistics
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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Measure of Central Tendency
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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Measure of Central Tendency
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: