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## Descriptive Statistics for one variable

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Title: Descriptive Statistics for one variable

1
Descriptive Statistics for one variable
2
Statistics has two major chapters
• Descriptive Statistics
• Inferential statistics

3
Statistics
• Descriptive Statistics
• Gives numerical and graphic procedures to
summarize a collection of data in a clear and
understandable way
• Inferential Statistics
• Provides procedures to draw inferences about a
population from a sample

4
Descriptive Measures
• Central Tendency measures. They are computed to
give a center around which the measurements in
the data are distributed.
• Variation or Variability measures. They describe
data spread or how far away the measurements
are from the center.
• Relative Standing measures. They describe the
relative position of specific measurements in the
data.

5
Measures of Central Tendency
• Mean
• Sum of all measurements divided by the number of
measurements.
• Median
• A number such that at most half of the
measurements are below it and at most half of the
measurements are above it.
• Mode
• The most frequent measurement in the data.

6
Example of Mean
• MEAN 40/10 4
• Notice that the sum of the deviations is 0.
• Notice that every single observation intervenes
in the computation of the mean.

7
Example of Median
• Median (45)/2 4.5
• Notice that only the two central values are used
in the computation.
• The median is not sensible to extreme values

8
Example of Mode
• In this case the data have tow modes
• 5 and 7
• Both measurements are repeated twice

9
Example of Mode
• Mode 3
• Notice that it is possible for a data not to have
any mode.

10
Variance (for a sample)
• Steps
• Compute each deviation
• Square each deviation
• Sum all the squares
• Divide by the data size (sample size) minus one
n-1

11
Example of Variance
• Variance 54/9 6
• It is a measure of spread.
• Notice that the larger the deviations (positive
or negative) the larger the variance

12
The standard deviation
• It is defines as the square root of the variance
• In the previous example
• Variance 6
• Standard deviation Square root of the variance
Square root of 6 2.45

13
Percentiles
• The p-the percentile is a number such that at
most p of the measurements are below it and at
most 100 p percent of the data are above it.
• Example, if in a certain data the 85th
percentile is 340 means that 15 of the
measurements in the data are above 340. It also
means that 85 of the measurements are below 340
• Notice that the median is the 50th percentile

14
For any data
• At least 75 of the measurements differ from the
mean less than twice the standard deviation.
• At least 89 of the measurements differ from the
mean less than three times the standard
deviation.
• Note This is a general property and it is
called Tchebichevs Rule At least 1-1/k2 of the
observation falls within k standard deviations
from the mean. It is true for every dataset.

15
Example of Tchebichevs Rule
• Suppose that for a certain data is
• Mean 20
• Standard deviation 3
• Then
• A least 75 of the measurements are between 14
and 26
• At least 89 of the measurements are between 11
and 29

16
Further Notes
• When the Mean is greater than the Median the data
distribution is skewed to the Right.
• When the Median is greater than the Mean the data
distribution is skewed to the Left.
• When Mean and Median are very close to each other
the data distribution is approximately symmetric.