Data observation and Descriptive Statistics

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Data observation and Descriptive Statistics
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Organizing Data

• Frequency distribution
• Table that contains all the scores along with the
  frequency (or number of times) the score occurs.
• Relative frequency proportion of the total
  observations included in each score.

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Frequency distribution
Amount f(frequency) rf(relative frequency)
0.00 2 0.125
0.13 1 0.0625
0.93 1 0.0625
1.00 1 0.0625
10.00 1 0.0625
32.00 1 0.0625
45.53 1 0.0625
56.00 1 0.0625
60.00 1 0.0625
63.25 1 0.0625
74.93 1 0.0625
80.00 1 0.0625
85.28 1 0.0625
115.35 1 0.0625
120.00 1 0.0625

n16 1.00
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Organizing data

• Class interval frequency distribution
• Scores are grouped into intervals and presented
  along with frequency of scores in each interval.
• Appears more organized, but does not show the
  exact scores within the interval.
• To calculate the range or width of the interval
• (Highest score lowest score) / of intervals
• Ex 120 0 / 5 24

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Class interval frequency distribution
Class interval f (frequency) rf ( relative frequency)
0-24 6 .375
25-48 2 .125
49-73 3 .1875
74-98 3 .1875
99-124 2 .125

n 16 1.00
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Graphs

• Bar graphs
• Data that are collected on a nominal scale.
• Qualitative variables or categorical variables.
• Each bar represents a separate (discrete)
  category, and therefore, do not touch.
• The bars on the x-axis can be placed in any
  order.

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Bar Graph
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Graphs

• Histograms
• To illustrate quantitative variables
• Scores represent changes in quantity.
• Bars touch each other and represent a variable
  with increasing values.
• The values of the variable being measured have a
  specific order and cannot be changed.

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Histogram
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Frequency polygon

• Line graph for quantitative variables
• Represents continuous data (time, age, weight)

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Frequency Polygon

• AGE
• 22.06
• 24.05
• 25.04
• 25.04
• 25.07
• 25.07
• 26.03
• 26.11
• 27.03
• 27.11
• 29.03
• 29.05
• 29.05
• 34
• 37.1
• 53

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Descriptive Statistics

• Numerical measures that describe
• Central tendency of distribution
• Width of distribution
• Shape of distribution

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Central tendency

• Describe the middleness of a data set
• Mean
• Median
• Mode

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Mean

• Arithmetic average
• Used for interval and ratio data
• Formula for population mean ( µ pronounced
  mu)
• µ ? X
• _____
• N
• Formulas for sample mean

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Mean
Amount f(frequency) rf(relative frequency)
0.00 2 0.125
0.13 1 0.0625
0.93 1 0.0625
1.00 1 0.0625
10.00 1 0.0625
32.00 1 0.0625
45.53 1 0.0625
56.00 1 0.0625
60.00 1 0.0625
63.25 1 0.0625
74.93 1 0.0625
80.00 1 0.0625
85.28 1 0.0625
115.35 1 0.0625
120.00 1 0.0625

46.53 n16 1
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Mean

• Not a good indicator of central tendency if
  distribution has extreme scores (high or low).
• High scores pull the mean higher
• Low scores pull the mean lower

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Median

• Middle score of a distribution once the scores
  are arranged in increasing or decreasing order.
• Used when the mean might not be a good indicator
  of central tendency.
• Used with ratio, interval and ordinal data.

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Median
0.00 0.00
0.13
0.93
1.00
10.00
32.00
45.53
56.00
60.00
63.25
74.93
80.00
85.28
115.35
120.00
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Mode

• The score that occurs in the distribution with
  the greatest frequency.
• Mode 0 no mode
• Mode 1 unimodal
• Mode 2 bimodal distribution
• Mode 3 trimodal distribution

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Mode
Amount f(frequency) rf(relative frequency)
0.00 2 0.125
0.13 1 0.0625
0.93 1 0.0625
1.00 1 0.0625
10.00 1 0.0625
32.00 1 0.0625
45.53 1 0.0625
56.00 1 0.0625
60.00 1 0.0625
63.25 1 0.0625
74.93 1 0.0625
80.00 1 0.0625
85.28 1 0.0625
115.35 1 0.0625
120.00 1 0.0625

46.53 n16 1
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Measures of Variability

• Range
• From the lowest to the highest score
• Variance
• Average square deviation from the mean
• Standard deviation
• Variation from the sample mean
• Square root of the variance

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Measures of Variability

• Indicate the degree to which the scores are
  clustered or spread out in a distribution.
• Ex Two distributions of teacher to student
  ratio.
• Which college has more variation?

College A College B
4 16
12 19
41 22
Sum 57 Sum 57
Mean 19 Mean 19
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Range

• The difference between the highest and lowest
  scores.
• Provides limited information about variation.
• Influenced by high and low scores.
• Does not inform about variations of scores not at
  the extremes.
• Examples
• Range X(highest) X (lowest)
• College A range 41- 4 37
• College B range 22-16 6

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Variance

• Limitations of range require a more precise way
  to measure variability.
• Deviation The degree to which the scores in a
  distribution vary from the mean.
• Typical measure of variability standard
  deviation (SD)
• Variance
• The first step in calculating standard deviation

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Variance

• X Number of therapy sessions each student
  attended.
• M 4.2

Deviation
Sum of deviations 0
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Variance

• In order to eliminate negative signs, we square
  the deviations.
• Sum the deviations sum of squares or SS

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Variance

• Take the average of the SS
• Ex SS 48.80
• SD2 S(X-M)2
• N
• That is the average of the squared deviations
  from the mean
• SD2 9.76

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Standard Deviation

• Standard deviation
• Typical amount that the scores vary or deviate
  from the sample mean
• SD S(X-M)2
• N
• That is, the square root of the variance
• Since we take the square root, this value is now
  more representative of the distribution of the
  scores.

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Standard Deviation

• X 1, 2, 4, 4, 10
• M 4.2
• SD 3.12 (standard deviation)
• SD2 9.76 (variance)
• Always ask yourself do these data (mean and SD)
  make sense based on the raw scores?

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Population Standard Deviation

• The average amount that the scores in a
  distribution vary from the mean.
• Population standard deviation
• (s pronounced sigma)

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Sample Standard Deviation

• Sample is a subset of the population.
• Use sample SD to estimate population SD.
• Because samples are smaller than populations,
  there may be less variability in a sample.
• To correct for this, we divide the sample by N
  1
• Increases the standard deviation of the sample.
• Provides a better estimate of population standard
  deviation.

• s ?( X - X ) ²
• _________
• N - 1

Unbiased Sample estimator standard deviation
Population standard deviation
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Sample Standard Deviation
X X - mean X - mean squared
0.00 -46.53 2,165.04
0.00 -46.53 2,165.04
0.13 -46.40 2,152.96
0.93 -45.60 2,079.36
1.00 -45.53 2,072.98
10.00 -36.53 1,334.44
32.00 -14.53 211.12
45.53 -1.00 1.00
56.00 9.47 89.68
60.00 13.47 181.44
63.25 16.72 279.56
74.93 28.40 806.56
80.00 33.47 1,120.24
85.28 38.75 1,501.56
115.35 68.82 4,736.19
120.00 73.47 5,397.84

46.53 N 16 SS 26,295.02
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Types of Distributions

• Refers to the shape of the distribution.
• 3 types
• Normal distribution
• Positively skewed distribution
• Negatively skewed distribution

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Normal Distribution

• Normal distributions Specific frequency
  distribution
• Bell shaped
• Symmetrical
• Unimodal
• Most distributions of variables found in nature
  (when samples are large) are normal
  distributions.

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Normal Distribution
Mean, media and mode are equal and located in the
center.
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Normal Distribution
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Skewed distributions

• When our data are not symmetrical
• Positively skewed distribution
• Negatively skewed distribution
• Memory hint skew is where the tail is also the
  tail looks like a skewer and it points to the
  skew (either positive or negative direction)

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Skewed Distributions
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Kurtosis

• Kurtosis - how flat or peaked a distribution is.
  
• Tall and skinny versus short and wide
• Mesokurtic normal
• Leptokurtic tall and thin
• Platykurtic short and fat (squatty like a
  platypus!)

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Kurtosis
leptokurtic
platykurtic
mesokurtic
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Skewness, Number of Modes, and Kurtosis in
Distribution of Housing Prices
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z - Scores

• In which country (US vs. England) is Homer
  Simpson considered overweight?
• How can we make this comparison?
• Need to convert weight in pounds and kilograms
  to a standardized scale.
• Z- scores allow for scores from different
  distributions to be compared under standardized
  conditions.
• The need for standardization
• Putting two different variables on the same scale
• z-score Transforming raw scores into
  standardized scores
• z (X - µ)
• s
• Tell us the number of standard deviations a score
  is from the mean.

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z- Scores

• Class 1 M 46.53 SD 41.87 X
  54.76
• Class 2 M 53.67 SD 18.23 X
  89.07
• In which class did I have more money in
  comparison to the distribution of the other
  students?
• Sample z-score z (X - M)
• s
• When we convert raw scores from different
  distributions to z-scores, these scores become
  part of the same z distribution and we
  can compare scores from different distributions.

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z Distribution

• Characteristics (regardless of the original
  distributions)
• z score at the mean equals 0
• Standard deviation equals 1

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z distribution of exam scores
M 70 s 10
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Standard normal distribution

• If a z-distribution is normal, then we refer to
  it as a standard normal distribution.
• Provides information about the proportion of
  scores that are higher or lower than any other
  score in the distribution.

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Standard Normal Curve Table

• Standard normal curve table (Appendix A)
• Statisticians provided the proportion of scores
  that fall between any two z-scores.
• What is the percentile rank of a z score of 1?
• Percentile rank proportion of scores at or
  below a given raw score.
• Ex SAT score 1350 M 1120 s 340
• 75th percentile

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Percentile Rank

• The percentage of scores that your score is
  higher than.
• 89th percentile rank for height
• You are taller than 89 of the students in the
  class. (you are tall!)
• Homer Simpson 4th percentile rank for
  intelligence.
• he is smarter than 4 of the population (or
  96 of the population is smarter than Homer).
• GRE score 88th percentile rank
• Reading scores of grammar school 18th percentile
  rank

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Review

• Data organization
• Frequency distribution, bar graph, histogram and
  frequency polygon.
• Descriptive statistics
• Central tendency middleness of a distribution
• Mean, median and mode
• Measures of variation the spread of a
  distribution
• Range, standard deviation
• Distributions can be normal or skewed (positively
  or negatively).
• Z- scores
• Method of transforming raw scores into standard
  scores for comparisons.
• Normal distribution mean z-score 0 and
  standard deviation 1
• Normal curve table shows the proportions of
  scores below the curve for a given z-score.