of 38 /38
Univariate Descriptive Statistics Dr. Shane Nordyke University of South Dakota This material is distributed under an Attribution-NonCommercial-ShareAlike 3.0 Unported Creative Commons License, the full details of which may be found online here: http://creativecommons.org/licenses/by-nc-sa/3.0/ . You may re-use, edit, or redistribute the content provided that the original source is cited, it is for non- commercial purposes, and provided it is distributed under a similar license. CC BY-NC-SA Nordyke 2010

Univariate Descriptive Statistics.pdf

Embed Size (px)

Citation preview

  • Univariate Descriptive Statistics

    Dr. Shane Nordyke

    University of South Dakota

    This material is distributed under an Attribution-NonCommercial-ShareAlike 3.0 Unported Creative Commons License, the full details of which may be found online here: http://creativecommons.org/licenses/by-nc-sa/3.0/ . You may re-use, edit, or redistribute the content provided that the original source is cited, it is for non-commercial purposes, and provided it is distributed under a similar license.

    CC BY-NC-SA Nordyke 2010

  • Why do we need descriptive statistics

    We use the label univariate descriptive statistics to refer to a variety of measures of center and variation that are useful for understanding the nature and distribution of a single variable.

    They can allow us to quickly understand a large amount of information about a single variable.

    They make data meaningful!

    CC BY-NC-SA Nordyke 2010

  • Making Data Meaningful

    Age of Volunteer 15 19

    22 17 39

    17 26

    CC BY-NC-SA Nordyke 2010

    A relatively small sample of the ages of volunteers at a local non-profit agency in the community.

    What does this list tell us about the age of volunteers in the agency?

  • Making Data Meaningful

    Age of Volunteer

    15 17 17 19 22 26 39

    CC BY-NC-SA Nordyke 2010

    Sorting the list can provide a starting place.

    What do we know now?

  • Making Data Meaningful

    CC BY-NC-SA Nordyke 2010

    What if the sample is larger?

    39 25 22 40 37 15 30 16 25 28

    16 31 50 46 30 15 25 20 17 22 43 27 42 43 17 16 33 26 31 30

    38 43 40 22 19 15 24 19 26 40 39 27 35 28 26 28 41 43 47 22

    36 41 25 38 25 36 38 38 18 45 16 30 40 21 16 48 48 46 30 31

    31 16 26 49 24 44 39 15 21 24 24 41 42 49 44 24 18 28 22 38

    22 47 44 20 31 24 24 27 34 33 17 49 33 44 27 43 49 16 23 25 35 34 20 26 29 44 17 42 43 29

    32 33 18 24 45 50 21 39 40 21 28 31 19 16 26 26 16 45 22 21

    47 15 39 49 33 29 40 20 18 37

    49 16 19 23 34 37 18 15 19 41

  • The Menu of Basic Descriptive Statistics

    Measures of central tendency

    Mean, Median, Mode, Midrange

    Measures of distribution

    Range, Min, Max, Percentiles

    Measures of Variation

    Standard Deviation, Variance, Coefficient of Variation

    CC BY-NC-SA Nordyke 2010

  • Some initial notation

    CC BY-NC-SA Nordyke 2010

    indicates the addition of a set of values

    y is the variable used to represent the individual data values

    n represents the number of values in a sample

    N represents the number of values in a population

  • Measures of Central Tendency - Mean

    The sample mean is the mathematical average of the data and is the measure of central tendency we use most often.

    CC BY-NC-SA Nordyke 2010

  • Measures of Central Tendency - Mean

    Sample Mean:

    = =1

    =155

    7

    = 22.14

    CC BY-NC-SA Nordyke 2010

    Observation #

    Age of Volunteer

    1 15 2 17 3 17 4 19 5 22 6 26 7 39

    155 The sum of all of the observations

    n = the number of observations

  • Measures of Central Tendency - Median

    The sample median is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude. If there isnt one value in the middle we take the average of the two middle values.

    The median is not affected by extreme values.

    CC BY-NC-SA Nordyke 2010

  • Measures of Central Tendency - Median

    CC BY-NC-SA Nordyke 2010

    =( )

    2 Median:

    Median is often denoted by which is pronounced y-tilde

  • Measures of Central Tendency - Median

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 39

    Sample ages are arranged in ascending order

    The middle value is the median. = 19

  • Measures of Central Tendency - Median

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 34 39

    If there are two values in the middle, we take the average of the two.

    =( )

    2

    =(19:22)

    2 = 20.5

  • Measures of Central Tendency - Median

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 34 99

    Note that the presence of an extreme value, doesnt change the median.

    =( )

    2

    =(19:22)

    2 = 20.5

  • Measures of Central Tendency - Mode

    The mode is the value that occurs most frequently.

    Not every sample has a distinct mode. Sometimes it is bimodal (two modes) or multimodal (three or more modes) or sometimes there is no mode at all.

    The mode is the only measure of central tendency we can use for nominal data.

    CC BY-NC-SA Nordyke 2010

  • Measures of Central Tendency - Mode

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 39

    17 is the only value that occurs more than once, so it is the value that occurs most

    frequently and the mode.

    Mode is often denoted with the symbol M

    M = 17

  • Measures of Central Tendency - Mode

    Blue Green Green Purple Purple Red Red Red Red Yellow Yellow Yellow

    CC BY-NC-SA Nordyke 2010

    M = Red

    20 29 33 33 34 41 41 42 43 45 45

    Multi modal

    1.1 2.3 4.1 5.3 4.3 6.7 8.2 8.3 8.7 8.9

    10.3

    No Mode

  • Measures of Central Tendency - Midrange

    The midrange, or middle of the range is the average of the highest and lowest values.

    There is no distinct symbol for the Midrange.

    CC BY-NC-SA Nordyke 2010

    Midrange=( : )

    2

  • Measures of Central Tendency - Midrange

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 39

    Midrange=( : )

    2

    Midrange=(15:39)

    2

    Midrange= 27

  • Comparing Measures of Central Tendency

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 39

    Mean = 22.14 Median = 19 Mode = 17 Midrange = 27

  • Comparing Measures of Center

    Measure of Center (Listed from most

    used to least used)

    Does it always exist?

    Does it take into account every

    value?

    Is it affected by extreme values?

    Mean Always Yes Yes

    Median Always No No

    Mode Might not exist, may have more than one

    No No

    Midrange Always No Yes

    CC BY-NC-SA Nordyke 2010

  • The Range

    The range of a sample is the difference between the highest value and the lowest value.

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 39

    In our example the Range = 39 15 or 24; there are 24 years between our youngest and oldest volunteers in the sample.

  • Measures of Variance

    Where measures of central tendency try to give us an idea of where the middle of the data lies, measures of variance (or variation) tell us about how the data is distributed around that center.

    Our three primary measures of variance are: Standard Deviation,

    Variance and

    Coefficient of Variation

    CC BY-NC-SA Nordyke 2010

  • Measures of Variance Standard Deviation

    Sample Standard Deviation: = (;)=1

    ;1

    2

    Population Standard Deviation: = (;)=1

    2

    CC BY-NC-SA Nordyke 2010

    The Standard Deviation is a measure of the variation of values around the mean.

  • Some Key Points for Understanding Standard Deviation

    The standard deviation is always positive.

    The standard deviation of a sample will always be in the same units as the observations in the sample.

    Extreme values or outliers can change the value of the standard deviation substantially.

    The size of the sample will affect the size of the standard deviation; as the sample size increases, the size of the standard deviation decreases.

    CC BY-NC-SA Nordyke 2010

  • Measures of Variance - Variance

    The variance of a sample is just the standard deviation of the sample squared.

    Sample Variance: 2 = (;)=1

    ;1

    Population Variance: 2 = (;)=1

    2

    CC BY-NC-SA Nordyke 2010

  • Standard Deviation and Variance Notation

    Sample Population

    s = standard deviation = standard deviation

    s2 = variance 2 = variance

    CC BY-NC-SA Nordyke 2010

  • Seeing Standard Deviations

    Once I figure out how to draw the curves, this well be a slide that shows the difference between a distribution with a small standard deviation (tall and narrow) and a large one (broad and flat).

    CC BY-NC-SA Nordyke 2010

  • Back to our example

    In our sample of volunteer ages, the mean was 22.14 years.

    We can calculate the standard deviation to better understand how the values or distributed around that mean.

    CC BY-NC-SA Nordyke 2010

    15 17 17 19 22 26 39

  • Back to our example

    Sample Standard Deviation: = (;)=1

    ;1

    2

    CC BY-NC-SA Nordyke 2010

    y (y-) (y-)2 15 22.14 -7.14 50.9796 17 22.14 -5.14 26.4196 17 22.14 -5.14 26.4196 19 22.14 -3.14 9.8596 22 22.14 -0.14 0.0196 26 22.14 3.86 14.8996 39 22.14 16.86 284.2596

    412.8572

  • Back to our example

    Sample Standard Deviation: = (;)=1

    ;1

    2

    = 412.86

    7 1

    CC BY-NC-SA Nordyke 2010

    = 8.3

  • Copyright 2004 Pearson Education,

    Inc.

    How are standard deviations helpful?

    The Empirical Rule

    When data sets have distributions that are approximately bell shaped, the following is true:

    About 68% of all values fall within 1 standard deviation of the mean

    About 95% of all values fall within 2 standard deviations of the mean

    About 99.7% of all values fall within 3 standard deviations of the mean

  • The Empirical Rule

    CC BY-NC-SA Nordyke 2010

    34% 34%

    68% of values fall within 1 standard deviation of the

    mean

  • The Empirical Rule

    CC BY-NC-SA Nordyke 2010

    34% 34%

    68% of values fall within 1 standard deviation of the

    mean

    95% of values fall within 2 standard deviations of the mean

    13.5% 13.5%

  • The Empirical Rule

    CC BY-NC-SA Nordyke 2010

    34% 34%

    68% of values fall within 1 standard deviation of the

    mean

    95% of values fall within 2 standard deviations of the mean

    99.7% of values fall within 3 standard deviations of the mean

    13.5% 13.5% 2.4% 2.4%

  • Measures of Center Coefficient of Variation

    The Coefficient of Variation (CV) is a measure of the standard deviation of a sample relative to its mean.

    CVs can be useful when you are comparing the standard deviations of variables that are in two different units.

    CC BY-NC-SA Nordyke 2010

  • Measures of Center Coefficient of Variation

    An example: You are comparing the heights and weights of fourth graders.

    CC BY-NC-SA Nordyke 2010

    Height = 52 S = 4

    Weight = 80 lbs. S = 10 lbs.

    Which variable has greater variance? How can we compare 4 to 10 lbs?

  • Measures of Center Coefficient of Variation

    CC BY-NC-SA Nordyke 2010

    Height = 52 S = 4

    CV =4

    52 * 100%

    CV = 8%

    CV =

    * 100%

    Weight = 80 lbs. S = 10 lbs.

    CV =10

    80 * 100%

    CV = 12.5%

    The standard deviation of height is 8% of the mean of height, where as the standard deviation of weight is 12.5% of the mean of weight, so there is greater variation in the weight of the fourth graders than in the height.