The Information School of the University of Washington LIS 570 Session 6.1 Univariate Data Analysis.

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LIS 570

Session 6.1Univariate Data Analysis

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LIS 570 Univariate Analysis

Mason; p. 2

Objectives: Have answers to the

following questions• Why is the normal distribution important for statistical analysis (the ones presented) to make sense?

• What is the logic behind inferential statistics?(On what theories is it based?)

• What is a Confidence Interval?• In what ways can we summarize

quantitative data?• What are some visualization techniques to

help us summarize and make sense of data?

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Mason; p. 3

Agenda• Exercise: understand “the

problem”• Vocabulary• Functions of statistics• When to use what type• Descriptive statistics• Inferential statistics

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Mason; p. 4

Why and What• Why know statistics?

– Informed consumer…

– Informed user…

– Informed professional…

– …

• What is a statistic?a descriptive summary (index) of a

sample

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Mason; p. 5

Sample and Population

SampleA set of observations,

instances, individuals drawn from a population, usually intended to represent the population in a study

Population (Universe)

The totality of things we are interested in (e.g., the population of all students at the UW)

SamplePopulation

Average = 4.5 Average = 4.55

statistic parameter

New

voca

bula

ry

A statistic is a characteristic of a sample, while the same characteristic, if descriptive of a population, is called a population parameter.

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Mason; p. 6

2 major functions of statistics

• Help us describe characteristics of sample– Descriptive statistics – Procedures to summarize, organize, and

simplify data• Help us describe characteristics of

population– Inferential statistics– Techniques for studying samples, and

then make generalizations about the population from which the samples were selected.*

* Source: Gravetter, F. J. and Wallnau, L. B. (2002). Essentials of Statistics for the Behavioral Sciences. 4th edition. Pacific Grove, CA: Wadsworth, p. 5

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Mason; p. 7

Vocabulary• Variable—characteristic which has more

than one value– e.g., Sex—male, female; hours of work/week—

anything from 0 – 168

– Independent variable (X)—manipulated by the researcher or believed to be the cause of…

– Dependent variable (Y)—variable observed to assess the effect of the manipulation, or changes depending on the independent variable

• Data—observations (measurements) taken on the units of analysis

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Mason; p. 8

Choosing the Statistical Technique*

Specific research question or hypothesis

Determine # of variables in question

Univariate analysis Bivariate analysis Multivariate analysis

Determine level of measurement of variables

Choose univariate method of analysis

Choose relevantdescriptive statistics

Choose relevantinferential statistics

* Source: De Vaus, D.A. (1991) Surveys in Social Research. Third edition. North Sydney, Australia: Allen & Unwin Pty Ltd., p133

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Mason; p. 9

What To Do with a Bunch of Numbers

• Organize the observations• Interested primarily in normality and deviations from

normality• Examine

– Central tendency

– Dispersion

– Shape of distribution

• Visualization aids– Frequency distribution (percentile) tables and charts – Histograms– Bar & pie charts (nominal data)– Frequency polygon– Cumulative percentage curve– Stem and leaf diagrams– Box plots

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Mason; p. 10

Frequency Distributions

• Ungrouped frequency distribution• A list of each of the values of the variable• The number of times and/or the percent

of times each value occurs

• Grouped frequency distribution • A table or graph • Shows frequencies or percent for ranges

of values

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Mason; p. 11

Frequency distributionsInclude in frequency distribution

tables:– Table number and title– Labels for the categories of the variables– Column headings– Total number of cases (N)– The number of missing cases– Source of the data– Footnotes to explain anomalies and notes


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Mason; p. 12

Grouped frequency distribution

Score Range

(Your value label)

Real Limits*

Frequencies (ƒ)

Cumulative frequencies

(Cf)

Percent (%)

Cumulative Percent

9-108.5 -

10.4999 3 20 15 100

7-86.5 -

8.4999 4 17 20 85

5-64.5 -

6.4999 7 13 35 65

3-42.5 -

4.4999 4 6 20 30

1-20.5 -

2.4999 2 2 10 10

Total (N) 20 100

Table 1—Example of grouped frequency distribution

Valid cases: 20 Missing cases: 0

Note 1: “Real limits” of a score extend from one-half of the smallest unit of measurement below the value of the score to one half unit above.

Note 2: Percent (%) = (ƒ /N) * 100, Cumulative % = (Cf/N) * 100

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Mason; p. 13

score interval

s ƒ

45-49 1

50-54 2

55-59 4

60-64 4

65-69 7

70-74 9

75-79 16

80-84 10

85-89 7

90-94 6

95-99 2

-The height of the bar corresponds to the frequency (ƒ)

-The width of the bar extends to the real limits of the score

-Used only on interval and ratio scales

-No space between bars (that’s a bar chart)

Histogram

47 52 57 62 67 72 77 82 87 92 970

5

10

15

20

Statistics exam scores

Fre

qu

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Mason; p. 14

What do graphs (histograms) show?

• Normality (normal distributions) [Why are normal distributions important?]

• Deviations from normality– Positive skewness– Negative skewness– Bimodality – And more…

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Mason; p. 15

Shapes of distributionNormal distribution:symmetrical Bell-shapedcurve

Positively skewed:tail on the right, cluster towards low end of the variable

Negatively skewed:tail on the left, cluster towards high-end of the variable

sym

metr

ical

Bimodality: A double peak

asym

metr

ical

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Mason; p. 16

• Central tendency is a single summary figure that ideally, is the most representative value of all values in the distribution.

• Used to describe “typical” or representative value Mean (arithmetic mean), m– Sum all the observations; divide by N: use for

interval variables when appropriate– Median: Value that divides the distribution so

that an equal number of values are above the median and an equal number below

– Mode: Value with the greatest frequency (uni-modal, bi-modal, etc.)

Central Tendency

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Mason; p. 17

Variability, dispersion, spread• Why do we care about anything

besides central tendency?

• Variability refers to spread or dispersion

• The extent to which a set of scores scatter about or cluster together

• Measures of variability– Range– Interquartile range– Sum-of-squares– Variance– Standard deviation– Kurtosis

Equal means, unequal variability

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Mason; p. 18

Kurtosis

Two distributions: the same mean & variance

Karl Pearson suggested names• Longer tailed: leptokurtic• Shorter tailed: platykurtic

http://members.aol.com/jeff570/k.html

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Mason; p. 19

Mode (Mo): most common value

• Best for nominal level data

• Cautions:– most common may not measure typicality– not sensitive to outliers (good and bad)– may be more than one mode– unstable from sample to sample

• Dispersion– variation ratio (v)

• % of people not in the modal category

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Mason; p. 20

Median (Mdn): Even split of sample

• For interval or ratio data, good for skewed distributions (mean would not be a good measure of central tendency)

• Minimal calculation (need to know frequencies)

• Reasonably insensitive to outliers (as long as there are only a few)

• Reasonably stable from sample to sample

• Example of ordinal variables– people are ranked from low to high (e.g., height)– median is the middle case– the median category is the one to which the middle person

belongs

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Mason; p. 21

Median– simple examples

– 1 2 3 4 5 6 7• Mdn = 4

– 1 2 3 5 6 7 9 13• Mdn = 5.5

by interpolation between 5 & 6 (5+6)/2 = 11/2 = 5.5

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Mason; p. 22

Dispersion• The nth percentile of a set of numbers

is a value such that n percent of the numbers fall below it and the rest fall above.– The median is the 50th percentile– The lower quartile is the 25th percentile– The upper quartile is the 75th percentile

• Summary of sample using 5 numbers: median, mean, variance, and extremes

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Mason; p. 23

Dispersion

Lowerquartile

Median Upperquartile

Top 25%Bottom 25%

Interquartile range

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Mason; p. 24

Boxplot

10864 12 14 16

Variable 1

Variable 2

Variable 3

Interquartile range (IQR)

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Mason; p. 25

Mean• Uses the actual numerical values of the

observations

• Most stable from sample to sample

• Most common measure of center

• Makes sense only for interval or ratio data

• Frequently computed for ordinal variables as well

• Not a good representation of central tendency for skewed samples

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Mason; p. 26

Mean--Dispersion• The standard deviation and variance measure

spread about the mean as centre.

• Deviation: distance and direction from the mean– Doesn’t work as a measure of variability because

adds up to zero (see next slide).

• Variance – mean of the squared deviation scores (of the

deviations of observations from the mean).

• Standard deviation – Conceptually: the typical distance of scores from the

mean– Technically: the square root of the variance

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Mason; p. 27

x = 6+7+5+3+4 = 25 = 55 5

– Variance (S2)• Calculate the mean for the variable• Take each observation and subtract the

mean from it• Square the result from the above• Add (sum) all the individual results• Divide by n

Example Data (6,7,5,3,4)

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Mason; p. 28

Observation x

Deviation x - x

Sq. deviation (x - x)2

6 6-5 = 1 1 7 7-5 = 2 4 5 5-5 = 0 0 3 3-5 = -2 4 4 4-5 = -1 1 Sum = 10

Variance (s2)

Variance = sum of the sq deviations = 10 = 2 number of observations 5

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Mason; p. 29

Standard deviation (s)

• Square root of the variance 2 = 1.4

• An average deviation of the observations from their mean

• Influenced by outliers

• Best used with symmetrical distributions

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Mason; p. 30

Summary• Descriptive statistics – univariate

analysis(central tendency, frequency distribution, dispersion)

• Determine if variable is nominal, ordinal or interval

• Nominal: frequency tables, mode• Ordinal

– Frequency tables (grouped frequency tables)– histogram– Median and five number summary– Mode

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Mason; p. 31

SummaryInterval

Determine whether the distribution is skewed or symmetrical

Compare median and mean

Use the mean and the standard deviation if the distribution is not markedly skewed; otherwise use five number summary (median, extremes, mid-quartile numbers)

Use the mode in addition if it adds anything

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Mason; p. 32

Abstract and Elevator Speech

20-30 second synopsis; intent: to elicit interest

• Who you are and what you are doing

• With whom• Where/How• Why: What you

hope to find, why the results may be important

100-300 words; elicit interest and summarize

• What type of study• How approached• When, where • Why: what you

hope to find, why the results may be important

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Mason; p. 33

Selecting analysis and statistical techniques*

Specific research question or hypothesis

Determine # of variables in question

Univariate analysis Bivariate analysis Multivariate analysis

Determine level of measurement of variables

Choose univariate method of analysis

Choose relevantdescriptive statistics

Choose relevantinferential statistics


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Mason; p. 34

Exercise—sampling distribution

• Coins, coins! • Probability of head or tails—50%• Each of you is a “sample” for this

activity.• Flip the coin 7 times, count the #

of times you get a “head”.

Live demo: http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

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Mason; p. 35

Why is normalit

y importan

t?

• Use proportions of the normal distribution to determine probabilities associated with any specific sample.

• Sampling Error• Standard Error (SE)—a way for defining and measuring sampling error

(exactly, how much error, on average, should exist between a sample mean and the unknown population mean, simply due to chance.

68%

95%100%

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Mason; p. 36

Standard Error of the mean

Standard error of the mean (Sm)

Sm = N

– Standard error is inversely related to square root of sample size

– To reduce standard error, increase sample size– Standard error is directly related to standard

deviation – When N = 1, standard error is equal to

standard deviation

Standard deviationTotal number in the sample

SS

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Mason; p. 37

Inferential statistics - univariate analysis

Interval estimates and interval variables• Estimation of sample mean accuracy—

based on random sampling and probability theory– Standardize the sample mean to estimate

population mean:t = sample mean – population mean

estimated SE

– Population mean = sample mean + t * (estimated SE)

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Mason; p. 38

Confidence IntervalUtilizes probability theory, assumes normal distribution

• 95% of the samples will fallwithin 1 to 2 standarddeviations from the population mean

• By the same token, for 95% of samples, the population mean will be within + or - 2 standard error units from the sample mean

• E.g., for C.I. 80%, first find the lower and upper t-values that bind 80% area of the distribution.

• Can state: with 80% confidence interval, the population mean is: sample mean + t (SE)

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Mason; p. 39

Standard Error(for nominal & ordinal

data)Variable must have only two

categories(may have to combine categories to

achieve this)

SB = PQ

NStandard error for binominal distribution

P = the % in one category of the variableQ = the % in the other category of the variable

Total number in the sample

The Information School of the University of Washington LIS 570 Session 6.1 Univariate Data Analysis.

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Transcript of The Information School of the University of Washington LIS 570 Session 6.1 Univariate Data Analysis.