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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.
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