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    4 Descriptive Statistics

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    NguyễnLanHương
    This document provides an overview of descriptive statistics including measures of central tendency, variation, and relative standing. It defines key terms like parameter, statistic, mean, median, mode, and variance. Examples are given to demonstrate how to calculate and interpret the mean, median, mode, variance, and standard deviation. It also discusses how adding or multiplying all values in a data set by a constant affects the mean, variance, and standard deviation.

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    4 Descriptive Statistics

    Uploaded by

    NguyễnLanHương
    338 views13 pages
    This document provides an overview of descriptive statistics including measures of central tendency, variation, and relative standing. It defines key terms like parameter, statistic, mean, median, mode, and variance. Examples are given to demonstrate how to calculate and interpret the mean, median, mode, variance, and standard deviation. It also discusses how adding or multiplying all values in a data set by a constant affects the mean, variance, and standard deviation.

    Original Description:

    Descriptive statistics: a parameter is a piece of information about the entire population

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    This document provides an overview of descriptive statistics including measures of central tendency, variation, and relative standing. It defines key terms like parameter, statistic, mean, median, mode, and variance. Examples are given to demonstrate how to calculate and interpret the mean, median, mode, variance, and standard deviation. It also discusses how adding or multiplying all values in a data set by a constant affects the mean, variance, and standard deviation.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    338 views13 pages

    4 Descriptive Statistics

    Uploaded by

    NguyễnLanHương
    This document provides an overview of descriptive statistics including measures of central tendency, variation, and relative standing. It defines key terms like parameter, statistic, mean, median, mode, and variance. Examples are given to demonstrate how to calculate and interpret the mean, median, mode, variance, and standard deviation. It also discusses how adding or multiplying all values in a data set by a constant affects the mean, variance, and standard deviation.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 13

    8/13/2012

    Descriptive statistics
    A parameter is a piece of information about the
    entire population. A statistic is our best guess
    for the parameter using only a sample.

    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.

    8/13/2012

    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.

    Mean

    8/13/2012

    Example of Mean
    Measurements
    x

    Deviation
    x - mean

    -1

    -3

    -2

    -4

    40

    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.

    Median

    8/13/2012

    Example of Median
    Measurements Measurements
    Ranked
    x
    x
    3
    0
    5
    1
    5
    2
    1
    3
    7
    4
    2
    5
    6
    5
    7
    6
    0
    7
    4
    7
    40
    40

    Median: (4+5)/2 =
    4.5
    Notice that only the
    two central values are
    used in the
    computation.
    The median is not
    sensible to extreme
    values

    8/13/2012

    Example of Mode
    Measurements
    x
    3
    5
    5
    1
    7
    2
    6
    7
    0
    4

    In this case the data have


    two modes:
    5 and 7
    Both measurements are
    repeated twice

    8/13/2012

    Variance

    xi x

    n 1

    Example of Variance
    Variance = 54/9 = 6
    Measurements Deviations
    x
    3
    5
    5
    1
    7
    2
    6
    7
    0
    4
    40

    x - mean
    -1
    1
    1
    -3
    3
    -2
    2
    3
    -4
    0
    0

    Square of
    deviations
    1
    1
    1
    9
    9
    4
    4
    9
    16
    0
    54

    It is a measure of
    spread.
    Notice that the larger the
    deviations (positive or
    negative) the larger the
    variance

    8/13/2012

    x x

    s s2

    n 1

    Measures of Location and Spread


    Discrete Outcomes
    Population

    Sample

    Measures of Location
    Mean

    x
    n

    Median

    Middle observation once all observations are ranked;


    average of two middle if the number of observations is even

    Mode

    Most often occurring value

    Measures of Spread

    Variance

    x x

    s2

    Standard Deviation
    Range

    Max Min

    n 1

    x x

    s s2

    n 1

    Max - Min

    8/13/2012

    Example-Calculation
    Data Set 1
    1
    3
    5
    7
    9

    Data Set 2
    1
    2
    5
    8
    9

    Data Set 3 Data Set 4


    3
    1
    5
    1
    7
    8
    9
    12
    11
    13

    Mean
    Median
    Mode
    Variance
    St. Dev.
    Range

    Example-Calculation

    Mean
    Median
    Mode
    Variance
    St. Dev.
    Range

    Data Set 1
    1
    3
    5
    7
    9

    Data Set 2
    1
    2
    5
    8
    9

    5
    5
    #N/A
    10
    3.162
    8

    5
    5
    #N/A
    12.5
    3.536
    8

    Data Set 3 Data Set 4


    3
    1
    5
    1
    7
    8
    9
    12
    11
    13

    7
    7
    #N/A
    10
    3.162
    8

    7
    8
    1
    33.5
    5.788
    12

    8/13/2012

    Some statistical rules


    applied to Variance
    What happens to mean, variance and
    standard deviation if we add subtract multiply
    or divide each number by a constant?

    The new data has the same


    spreadoutness but a new mean
    So if you add a constant C to each
    Xi, then

    8/13/2012

    Example temperature conversions to get


    from Fahrenheit to Celsius use

    So we will convert each of the


    temperatures from farenheit to Celsius
    remember

    10

    8/13/2012

    Which value is father from the mean, 27F


    or -1c? Since these values are in different
    units, we need a standardized way to
    compare them.

    11

    8/13/2012

    12

    8/13/2012

    What happens to mean, variance and


    standard deviation if we add subtract
    multiply or divide each number by a
    constant?

    13

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