Action Research Measurement Scales and Descriptive Statistics

1
Action ResearchMeasurement Scales and
Descriptive Statistics

• INFO 515
• Glenn Booker

2
Measurement Needs

• Need a long set of measurements for one project,
  and/or many projects to examine statistical
  trends
• Could use measurements to test specific
  hypotheses
• Other realistic uses of measurement are to help
  make decisions and track progress
• Need scales to make measurements!

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Measurement Scales

• There are four types of measurement scales
• Nominal
• Ordinal
• Interval
• Ratio
• Completely optional mnemonic to remember the
  sequence, I think of NOIR like in the
  expression film noir (noir is French for
  black)

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Nominal Scale

• A nominal (name) scale groups or classifies
  things into categories, which
• Must be jointly exhaustive (cover everything)
• Must be mutually exclusive (one thing cant be
  in two categories at once)
• Are in any sequence (none better or worse)
• So a nominal variable is putting things into
  buckets which have no inherant order to them

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Nominal Scale

• Examples include
• Gender (though some would dispute limitations of
  only male/female categories)
• Dewey decimal system
• The Library of Congress system
• Academic majors
• Makes of stuff (cars, computers, etc.)
• Parts of a system

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Ordinal Scale

• This measurement ranks things in order
• Sequence is important, but the intervals between
  ranks is not defined numerically
• Rank is relative, such as greater than or less
  than
• E.g. letter grades, urgency of problems, class
  rank, inspection ratings
• So now the buckets were using have some sense or
  order or direction

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Interval Scale

• An interval scale measures quantitative
  differences, not just relative
• Addition and subtraction are allowed
• E.g. common temperature scales (F or C), a
  single date (Feb 15, 1999), maybe IQ scores
• Let me know if you find any more examples
• A zero point, if any, is arbitrary (90 F is
  not six times hotter than 15 F!)

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Ratio Scale

• A ratio scale is an interval scale with a
  non-arbitrary zero point
• Allows division and multiplication
• The best type of scale to use, if possible
• E.g. defect rates for software, test scores,
  absolute temperature (Kelvin or Rankine), the
  number or count of almost anything, size, speed,
  length,

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Summary of Scales

• Nominal
• names different categories, not ordered, not
  ranked Male, Female, Republican, Catholic..
• Ordinal
• Categories are ordered Low, High, Sometimes,
  Never,
• Interval
• Fixed intervals, no absolute zero IQ,
  Temperature
• Ratio
• Fixed intervals with an absolute zero point Age,
  Income, Years of Schooling, Hours/Week, Weight
• Age could be measured as ratio (years), ordinal
  (young, middle, old), or nominal (baby boomer,
  gen X)
• Scale of measurement affects (may determine) type
  of statistics that you can use to analyze the data

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Scale Hierarchy

• Measurement scales are hierarchicalratio
  (best) / interval / ordinal / nominal
• Lower level scales can always be derived from
  data which uses a higher scale
• E.g. defect rates (a ratio scale) could be
  converted to High, Medium, Low or Acceptable,
  Not Acceptable (ordinal scales)

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Reexamine Central Tendencies

• If data are nominal, only the mode is meaningful
• If data are ordinal, both median and mode may be
  used
• If data are ratio or interval (called scale in
  SPSS), you may use mean, median, and mode

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Reexamine Variables

• Discrete variables use counting units or specific
  categories
• Example makes of cars, grades,
• Use Nominal or Ordinal scales
• Continuous Integer or Real Measurements
• Example IQ Test scores, length of a table, your
  weight, etc.
• Use Ratio or Interval scales

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Refine Research Types

• Qualitative Research tends to use Nominal and/or
  Ordinal scale variables
• Quantitative Research tends to use Interval
  and/or Ratio scale variables

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Frequency Distributions

• Frequency distributions describe how many times
  each value occurs in a data set
• They are useful for understanding the
  characteristics of a data set
• Frequencies are the count of how many times each
  possible value appears for a variable (gender
  male, or operating system Windows 2000)

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Frequency Distributions

• They are most useful when there is a fixed and
  relatively small number of options for that
  variable
• Theyre harder to use for variables which are
  numbers (either real or integer) unless there are
  only a few specific options allowed (e.g. test
  responses 1 to 5 for a multiple choice question)

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Generating Frequency Distributions

• Select the command Analyze / Descriptive
  Statistics / Frequencies
• Select one or more Variable(s)
• Note that the Frequency (count) and percent are
  included by default other outputs may be
  selected under the Statistics... button
• A bar chart can be generated as well using the
  Charts button see another way later

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Sample Frequency Output
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Analysis of Frequency Output

• The first, unlabeled column has the values of
  data here, it first lists all Valid values
  (there are no Invalid ones, or it would show
  those too)
• The Frequency column is how many times that value
  appears in the data set
• The Percent column is the percent of cases with
  that value in the fourth row, the value 15
  appears 116 times, which is 24.5 of the 474
  total cases (116/474100 24.5)

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Analysis of Frequency Output

• The Valid Percent column divides each Frequency
  by the total number of Valid cases ( Percent
  column if all cases valid)
• The Cumulative Percent adds up the Valid Percent
  values going down the rows so the first entry is
  the Valid Percent for first row, the second entry
  is from 11.2 40.1 51.3, next is 51.3 1.3
  52.5 and so on

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Generating Frequency Graphs

• Frequency is often shown using a bar graph
• Bar graphs help make small amounts of data more
  visible
• To generate a frequency graph alone
• Click on the Charts menu and select Bar
• Leave the Simple graph selected, and leave
  Summaries are for groups of cases selected
  click the Define button

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Generating Frequency Graphs

• Let the Bars Represent remain N of cases
• Click on variable Educational Level (years) and
  move it into the Category Axis field
• Click OK
• You should get the graph on the next
  slide.Notice that the text below the X axis is
  the Label for the Category Axis.

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Sample Frequency Output
Notice that the exact same graph can be generated
from Frequencies, or just as a bar graph
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Frequency Distributions

• A frequency distribution is a tabulation that
  indicates the number of times a score or group of
  scores occurs
• Bar charts best used to graph frequency of
  nominal ordinal data
• Histograms best used to display shape of interval
  ratio data

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Frequency Distribution Example
SPSS for Windows, Student Version
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Basic Measures - Ratio

• Used for two exclusive populations (every case
  fits into one OR the other)
• Ratio ( of testers) / ( of developers)
• E.g. tester to developer ratio is 14

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Proportions and Fractions

• Used for multiple (gt 2) populations
• Proportion (Number of this population)
  / (Total number of all populations)
• Sum of all proportions equals unity (one)
• E.g. survey results
• Proportions are based on integer units
• Fractions are based on real numbered units

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Percentage

• A proportion or fraction multiplied by 100
  becomes a percentage
• Only report percentages when N (total population
  measured) is above 30 to 50 and always provide
  N for completeness
• Why? Otherwise a percentage will imply more
  accuracy than the data supports
• If 2 out of 3 people like something, its
  misleading to report that 66.667 favor it

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Percents

• Percent the percentage of cases having a
  particular value.
• Raw percent divide the frequency of the value
  by the total number of cases (including missing
  values)
• Valid percent calculated as above but excluding
  missing values

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Percent Change

• The percent increase in a measurement is the new
  value, minus the old one, divided by the old
  value negative means decrease increase (new
  - old) / old
• The percent change is the absolute value of the
  percent increase or decrease change
  increase

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Percent Increase

• Later Value Earlier Value Earlier Value
• So if a collection goes from 50,000 volumes in
  1965 to 150,000 in 1975, the percent increase
  is
• 150,000-50,000 2 200 50,000
• Always divide by where you started

Carpenter and Vasu, (1978)
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Percentiles

• A percentile is the point in a distribution at or
  below a given percentage of scores.
• The median is the 50 percentile
• Think of the SAT scores - what percentile were
  you for verbal, math, etc. - means what percent
  of people did worse than you

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Rate

• Rate conveys the change in a measurement, such as
  over time, dx/dt. Rate ( observed events) / (
  of opportunities)constant
• Rate requires exposure to the risk being measured
• E.g. defects per KSLOC (1000 lines of code) (
  defects)/( of KSLOC)1000

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Exponential Notation

• You might see output of the form 2.78E-12
• The E means times ten to the power of
• This is 2.78 10-12 (2.7810-12)
• A negative exponent, e.g. 12, makes it a very
  small number
• 10-12 0.000000000001
• 1012 1,000,000,000,000
• The leading number, here 2.78, controls whether
  it is a positive or negative number

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Exponential Notation
51012 (a positive number gtgt1)
Pos.
510-12 (a positive number ltlt1)
0
-510-12 (a negative number ltlt1)
Neg.
-51012 (a negative number gtgt1)
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Precision

• Keep your final output to a consistent level of
  precision (significant digits)
• Dont report one value as 12 and another as
  11.86257523454574123
• Pick a level of precision to match the accuracy
  of your inputs (or one digit more), and make sure
  everything is reported that way consistently
  (e.g. 12.0 and 11.9)

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Data Analysis

• Raw data is collected, such as the dates a
  particular problem was reported and closed
• Refined data is extracted from raw data, e.g. the
  time it took a problem to be resolved
• Derived data is produced by analyzing refined
  data, such as the average time to resolve problems

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Descriptive Statistics

• Descriptive statistics describes the key
  characteristics of one set of data (univariate)
• Mean, median, mode, range (see also last week)
• Standard deviation, variance
• Skewness
• Kurtosis
• Coefficient of variation

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Mean

• A.k.a. Average Score
• The mean is the arithmetic average of the scores
  in a distribution
• Add all of the scores
• Divide by the total number of scores
• The mean is greatly influenced by extreme scores
  they pull it off center

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Mean Calculation
HOLDINGS IN 7 DIFFERENT LIBRARIES X Mean
?X N 7400 6500 39200
5600 6200 7 5900
5100 4300 Here, sum every data value 3800 ?
X 39200
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Mean with a Frequency Distribution
X (IQ) FFreq FX FX 140 2 280 135 1 135 1
32 2 264 130 1 130 128 1 128 126 1 126 125
4 500 123 1 123 120 4 480 110 3 330 101
1 101 21 2597 Mean
?FX 2597 123.67 124 (round off)
N 21 N SF
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Central Tendency Example
Staff Salaries 4100 6000 6000 Mode
6000 6000 8000 Median 9 1 5th
value 8000 9000
2 10000 11000 Mean ?X 80100
8900 20000 N 9
Carpenter and Vasu, (1978)
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Handling Extreme Values

• In cases where you have an extreme value (high or
  low) in a distribution, it is helpful to report
  both the median and the mean
• Reporting both values gives some indication
  (through comparison) of a skewed distribution

43
Measures of Variation

• Measures which indicate the variation, or spread
  of scores in a distribution
• Range (see last week)
• Variance
• Standard Deviation

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Standard Deviation, Variance

• Standard deviation is the average amount the data
  differs from the mean (average)SD ?( S
  (Xi-X)2 / (N-1) )SD ?( Variance )
• Variance is the standard deviation
  squaredVariance S (Xi-X)2 / (N-1)
• per ISO 3534-1, para 2.33 and 2.34

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Standard Deviation

• The standard deviation is the square root of the
  variance. It is expressed in the same units as
  the original data.
• Since the variance was expressed squared units
  it doesnt make much practical sense. For
  example, what are squared books or squared
  man-hours?

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Computing the VarianceS2 ?(X Mean)2
N

• 1. Subtract the mean from each score
• 2. Square the result
• 3. Sum the squares for all data points
• 4. Divide by the N of cases

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Divide by N or N-1???

• Youll see different formulas for variance and
  standard deviation some divide by N, some by
  N-1 (e.g. slides 43 and 45) why?
• If your data covers the entire population (you
  have all of the possible data to analyze), then
  divide by N
• If your data covers a sample from the population,
  divide by N-1

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Standard Deviation for Freq Dist.
X F FX X2 FX2 17 2 34 289 578 16 4 64 256
1024 14 5 70 196 980 10 2 20 100 200 9 3 27
81 243 6 1 6 36 36 221
3061 s v (?FX2 (?FX)2/N) v
(3061- (221)2/17) N
17 v ((3061- 2873)/17) 3.3 Notice
that FX2 is F(X2), not (FX)2
Standard Deviation of Bookmobile Distribution
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Std Dev Reflects Consistency
Distance from Target
Frequency In Meters Battery A
Battery B 200 2 0 150 4 1
100 5 5 50 7
10 0 9 13
-50 7
10 -100 5 5 -150 4 1 -200 2
0 Mean 0 Mean 0 Standard D.
Standard D. 102.74 65.83
Runyon and Haber (1984)
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Standard Deviation vs. Std. Error

• To be precise, the standard error is the standard
  deviation of a statistic used to estimate a
  population parameter per ISO 3534-1, para 2.56
  and 2.50
• So standard error pertains to sample data, while
  standard deviation should describe the entire
  population
• We often use them interchangeably ?

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Skewness

• Skewness is a measure of the asymmetry of a
  distribution.
• The normal distribution is symmetric, and has a
  skewness value of zero.
• A distribution with a significant positive
  skewness has a long right tail
• Positive skewness means the mean and median are
  more positive than the mode (the peak of the
  distribution)
• Negative skewness has a long left tail.

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Skewness

• As a rough guide, a skewness magnitude more than
  two (gt2 or lt-2) is taken to indicate a
  significant departure from symmetry

From www.riskglossary.com
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Kurtosis

• Kurtosis is a measure of the extent to which data
  clusters around a central point
• For a normal distribution, the value of the
  kurtosis is 3
• The kurtosis excess ( kurtosis-3) is zero for a
  normal distribution
• Positive kurtosis excess indicates that the data
  have longer tails than normal
• Negative kurtosis excess indicates the data have
  shorter tails

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Kurtosis
tail
The curve on the right has higher kurtosis than
the curve on the left. It is more peaked at the
center, and it has fatter tails. If a
distributions kurtosis is greater than 3, it is
said to be leptokurtic (sharp peak). If its
kurtosis is less than 3, it is said to be
platykurtic (flat peak). They might have equal
standard deviation. Mesokurtic is the normal
curve, which has kurtosis 3.
From www.riskglossary.com
55
Skewness Kurtosis Example

• From the Employee data set, use Analyze /
  Descriptive Statistics / Descriptives, select the
  salary variable
• Under Options, select Skewness and Kurtosis
• Skewness is 2.125, so there is significant
  positive skewness to the data
• Kurtosis is 5.378, so the data is leptokurtic

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Coefficient of Variation

• The coefficient of variation (CV) is the ratio of
  the standard deviation to the meanCV s/m
  per ISO 3534-1, para 2.35
• Smaller CV means the more representative the mean
  is for the total distribution
• Can compare means and standard deviations of two
  different populations
• Higher CV means more variability

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Coefficient of Variation

• Divide the standard deviation by the mean to get
  CV. CV s/m
• The smaller the decimal fraction this produces,
  the more representative is the mean for the total
  distribution
• The larger the decimal fraction, the worse job
  the mean does of giving us a true picture of the
  distribution

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Generating a Histogram

• Frequency graphs can be generated for variables
  which have many integer or real values (e.g.
  salary), by using a histogram
• A histogram shows how many data points fall into
  various ranges of values
• The closest normal curve can be shown for
  comparison

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Generating a Histogram

• The ¾ rule is helpful for histograms
• The tallest bar should be ¾ of the height of the
  Y axis
• Be sure to label X and Y axes appropriately
• The each bar shows how many data points fall
  within a range of X axis values
• See How to Lie with Statistics, by Darrell Huff

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Histogram of Salary
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Another Note on Histograms

• SPSS will define its own bar widths for a
  histogram, e.g. how wide the range of salary
  values is for each bar
• Later in the course, well look at how you can
  define your own variables to make predefined
  histograms bars

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Pie Chart Histogram

• A histogram can also be made in the shape of a
  pie
• This should be limited to variables with a small
  number of possible values

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A bad pie chart histogram
(I had to include this one just because its
colorful)
64
This is a better example
This visually implies the percentages of data in
each value.
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Bookmobile Data
Bookmobile examples taken from Carpenter and
Vasu, (1978) Same data as used on slides 48 66.
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Bookmobile Distributions
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HISTOGRAM OF BOOKMOBILE STOPS
F
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Normalizing Data

• Some data sets are not very close to a normal
  distribution
• Sometimes it helps to transform the independent
  variable by applying a math function to it, such
  as looking at log(x) (the logarithm of each x
  value) instead of just x

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Normalizing Data

• In SPSS this can be done by defining a new
  variable, such as log_x
• Then use Transform / Compute to calculate log_x
  LG10(x) assuming that x is the original
  variable
• Then generate a histogram showing the normal
  curve, to see if log_x is closer to a normal
  distribution

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Normalizing Data

• Who cares if we have a normal distribution?
• Many tests in statistics can only be applied to a
  variable which has a normal distribution so
  its worth our while to transform the variable