STATISTICS - SUMMARY
1. Functions of statistics
a. description: summarize a set of data
b. inference: make generalizations from sample to population. parameter estimates, hypothesis tests.
2. Types of statistics
i. descriptive statistics: describe a set of data
a. central tendency: mean, median (order statistics), mode.
b. dispersion: range, variance & standard deviation,
c. Others: shape -skewness, kutosis.
d. EDA procedures (exploratory data analysis).
Stem & leaf display: ordered array, freq distrib. & histogram all in one.
Box and Whisker plot: Five number summary-minimum,Q1, median, Q3, and
maximum.
Resistant statistics: trimmed and winsorized means,midhinge, interquartile deviation.
ii. inferential statistics: make inferences from samples to populations.
iii. Parameteric vs non-parametric statistics
a. parametric : generally assume interval scale measurements and normally distributed
variables.
b. nonparametric (distribution free statistics) : generally weaker assumptions: ordinal or
nominal measurements, don't specify the exact form of distribution.
3.Steps in hypothesis testing.
1. Make assumptions & choose the appropriate statistic. Check measurement scale of variables.
2. State null hypothesis; and the alternative
3. Select a confidence level for the test. Determine the critical region - values of the statistic for which
you will reject the null hypothesis.
4. Calculate the statistic.
5. Reject or fail to reject null hypothesis.
6. Interpret results.
Type I error: rejecting null hypothesis when it is true.
Prob of Type I error is 1-confidence level.
Type II error: failing to reject null hypothesis when it is false.
Power of a test = 1-prob of a type II error.
4. Null hypotheses for simple bivariate tests.
a. Pearson Correlation
rxy =0.
b. T-Test
x =y
c. One Way ANOVA
M1=M2=M3=...=Mn
d. Chi square : No relationship between x and y. Formally, this is captured by the "expected
table", which assumes cells in the X-Y table can be generated completely from row and
column totals.
5. EXAMPLES OF T-TEST AND CHI SQUARE
(1) T-TEST. Tests for differences in means (or percentages) across two subgroups. Null hypothesis is mean
of Group 1 = mean of group 2. This test assumes interval scale measure of dependent variable (the one you
compute means for) and that the distribution in the population is normal. The generalization to more than
two groups is called a one way analysis of variance and the null hypothesis is that all the subgroup means
are identical. These are parametric statistics since they assume interval scale and normality.
(2) Chi square is a nonparametric statistic to test if there is a relationship in a contingency table, i.e. Is the
row variable related to the column variable? Is there any discernible pattern in the table? Can we predict the
column variable Y if we know the row variable X?
The Chi square statistic is calculated by comparing the observed table from the sample, with an "expected"
table derived under the null hypothesis of no relationship. If Fo denotes a cell in the observed table and Fe a
corresponding cell in expected table, then
2
Chi square (  ) =
2
 (Fo -Fe) /Fe
cells
The cells in the expected table are computed from the row (nr ) and column (nc ) totals for the sample as
follows:
Fe =nr nc / n
.
CHI SQUARE TEST EXAMPLE: Suppose a sample (n=100) from student population yields the following
observed table of frequencies:
Male
GENDER
Female
Total
20
30
50
40
10
50
60
40
100
IM-USE
Yes
No
Total
EXPECTED TABLE UNDER NULL HYPOTHESIS (NO RELATIONSHIP)
Male
GENDER
Female
Total
30
20
50
30
20
50
60
40
100
IM-USE
Yes
No
Total
2
2
2
2
2
 = (20-30) /30 + (40-30) /30 + (30-20) /20 + (10-20) /20
100/30 + 100/30 + 100/20 +100/20 = 13.67
Chi square tables report the probability of getting a Chi square
value this high for a particular random sample, given that there is
no relationship in the population. If doing the test by hand, you
would look up the probability in a table. There are different Chi
square tables depending on the number of cells in the table.
Determine the number of degrees of freedom for the table as
(rows-1) X (columns -1). In this case it is (2-1)*(2-1)=1. The
probability of obtaining a Chi square of 13.67 given no relationship is less than .001. (The last entry in my
table gives 10.83 as the chi square value corresponding to a probability of .001, so 13.67 would have a
smaller probability).
If using a computer package, it will normally report both the Chi square and the probability or significance
level corresponding to this value. In testing your null hypothesis, REJECT if the reported probability is less
than .05 (or whatever confidence level you have chosen). FAIL TO REJECT if the probability is greater
than .05.
For the above example : REVIEW OF STEPS IN HYPOTHESIS TESTING:
(1) Nominal level variables, so we used Chi square.
(2) State null hypothesis. No relationship between gender and IM-USE
2
(3) Choose confidence level. 95%, so alpha = .05, critical region is  > 3.84 (see App E -p. A-28).
2
(4) Draw sample and calculate the statistic;  = 13.67
(5). 13.67 > 3.84, so inside critical region, REJECT null hypothesis. Alternatively, SIG= .001 on computer
printout, .001<.05 so REJECT null hypothesis.
Note we could have rejected null hypothesis at .001 level here.
WHAT HAVE WE DONE? We have used probability theory to determine the likelihood of obtaining a
contingency table with a Chi square of 13.67 or greater given that there is no relationship between gender
and IMUSE. If there is no relationship (null hypothesis is true), obtaining a table that deviates as much as
the observed table does from the expected table would be very rare - a chance of less than one in 1000. We
therefore assume we didn't happen to get this rare sample, but instead our null hypothesis must be false.
Thus we conclude there is a relationship between gender and IMUSE.
The test doesn't tell us what the relationship is, but we can inspect the observed table to find out. Calculate
row or column percents and inspect these. Percents below are row percents obtained by dividing each entry
on a row by the row total.
Row percents:
Male
GENDER
Female
Total
.33
.75
.50
.67
.25
.50
1.00
1.00
1.00
IM-USE
Yes
No
Total
To find the "pattern" in table, compare row percents for each row with the "Totals" at bottom. Thus, half of
sample are men, whereas only a third of IMusers are male and three quarters of nonusers are male.
Conclusion - men are less likely to use IM.
-------------------------------------------------------------Column Percents: Divide entries in each column by column total.
GENDER
Male
Female
Total
IM-USE
Yes
.40
.80
.60
No
.60
.20
.40
Total
1.00
1.00
1.00
PATTERN: 40% of males use IM, compared to 80% of women. Conclude women more likely to use IM.
Note in this case the column percents provide a clearer description of the pattern than row percents.
6. BRIEF NOTES AND SAMPLE PROBLEMS
a. Measures of strength of a relationship vs a statistical test of a hypothesis. There are a number of
statistics that measure how strong a relationship is, say between variable X and variable Y. These include
parametric statistics like the Pearson Correlation coefficient, rank order correlation measures for ordinal
data (Spearman's rho and Kendall's tau), and a host of non-parametric measures including Cramer's V, phi,
Yule's Q, lambda, gamma, and others. DO NOT confuse a measure of association with a test of a
hypothesis. The Chi square statistic tests a particular hypothesis. It tells you little about how strong the
relationship is, only whether you can reject a hypothesis of no relationship based upon the evidence in your
sample. The problem is that the size of Chi square depends on strength of relationships as well as sample
size and number of cells. There are measures of association based on chi square that control for the number
of cells in table and sample size. Correlation coefficients from a sample tell how strong the relationship is in
the sample, not whether you can generalize this to the population. There is a test of whether a correlation
coefficient is significantly different from zero that evaluates generalizability from the sample correlation to
the population correlation. This tests the null hypothesis that the correlation in the population is zero.
b. Statistical significance versus practical significance. Hypothesis tests merely test how confidently we
can generalize from what was found in the sample to the population we have sampled from. It assumes
random sampling-thus, you cannot do statistical hypothesis tests from a non-probability sample or a census.
The larger the sample, the easier it is to generalize to the population. For very large sample sizes, virtually
ALL hypothesized relationships are statistically significant. For very small samples, only very strong
relationships will be statistically significant. What is practically significant is a quite different matter from
what is statistically significant. Check to see how large the differences really are to judge practical
significance, i.e. does the difference make a difference?.
c. SOME SAMPLE/SIMPLE STATISTICAL PROBLEMS:
1. Calculate mean, median, standard deviation, variance from a set of data.
2. Compute Z scores to find areas under normal distribution.
3. Find an alpha (95%) percent confidence interval for the mean.(alpha given). Must estimate the standard
error of mean, 95% CI = 2 S.E.'s either side of mean.
4. Given a confidence level, accuracy desired, and estimate of variance, determine the required sample size
for a survey. n=Z22/ e2 , where Z is number of standard errors assoc. with confidence level,  is an
estimate of standard deviation of variable in the population, and e is size of error you can tolerate.
5. Similar problems for proportions rather than means. Simply replace standard deviation by sqrt(p(1-p)),
the standard deviation of a binomial distribution with probability p. n=Z 2p(1-p)/ e2
6.Chi square test of relationship between two nominal scaled variables.
7. Brief Summary of Multivariate Analysis Methods. SPSS procedure in Capitals.
1. Linear Regression: Estimate a linear relationship between a dependent variable and a set of independent
variables. All must be interval scale or dichotomous (dummy variables).(See Babbie p 437, T&H, p
619, Also JLR 15(4). Examples: estimating participation in recreation activities, cost functions,
spending. REGRESSION.
2. Non-linear models : Similar to above except for the functional form of the relationship. Gravity models,
logit models, and some time series models are examples. (See Stynes & Peterson JLR 16(4) for logit,
Ewing Leisure Sciences 3(1) for gravity. Examples: Similar to above when relationships are non-linear.
Gravity models widely used in trip generation and distribution models. Logit models in predicting
choices. NLR
3. Cluster analysis : A host of different methods for grouping objects based upon their similarity across
several variables. (See Romesburg JLR 11(2) & book review same issue.) Examples: Used frequently
to form market segments or otherwise group cases. See Michigan Ski Market Segmentation Bulletin
#391 for a good example.CLUSTER QUICK CLUSTER
4. Factor analysis. Method for reducing a large number of variables into a smaller number of independent
(orthogonal) dimensions or factors. (See Kass & Tinsley JLR 11(2); Babbie p 444, T&H pp 627).
Examples: Used in theory development (e.g. What are the underlying dimensions of leisure attitudes?)
and data reduction (reduce number of independent variables to smaller set). FACTOR.
5. Discriminant analysis: Predicts group membership using linear "discriminant" functions. This is a
variant of linear regression suited to predicting a nominal dependent variable. (See JLR 15(4) ; T&H
pp 625). Examples: Predict whether an individual will buy a sail, power, or pontoon boat based upon
demographics and socioeconomics. DISCRIMINANT
6. Analysis of Variance (ANOVA): To identify sources of variation in a dependent variable across one or
more independent variables. Tests null hypothesis of no difference in means of dependent variable for
three or more subgroups (levels or categories of independent variable). The basic statistical analysis
technique for experimental designs. (See T&H pp 573, 598). Multivariate analysis of variance
(MANOVA) is the extension to more complex designs. (See JLR 15(4)). ANOVA, MANOVA.
7. Multidimensional scaling (MDS): Refers to a number of methods for forming scales and identifying the
structure (dimensions) of attitudes. Differ from factor analysis in employing non-linear methods. MDS
can be based on measures of similarities between objects. Applic in recreation & tourism- mapping
images of parks or travel destinations. Identifying dimensions of leisure attitudes.RELIABILITY (See
T&H pp 376.)
8.Others: Path analysis (LISREL) (Babbie p. 441), canonical correlation, conjoint analysis (See T& H 359,
App C), multiple classification analysis, time series analysis (Babbie p. 443), log linear analysis
(LOGLINEAR HILOGLINEAR) linear programming, simulation.
Guidance to Statistical Tests - Hypothesis Tests
Nominal x Nominal
Chi Square : Test null hypothesis of no relationship between two nominal scale variables . Need minimum of
5 cases per cell in your table, so don’t run with variables that have too many categories (recode if neccessary
to collapse categories) . Use the Pearson Chi Square statistic in SPSS.
Nominal x Interval
T-Test : Tests null hypothesis of no difference in means between two groups. Independent variable divides
population into two groups, estimate means for each group on the dependent variable (interval scale).
One Sample T-test tests if mean of a single group differs from some constant.
Independent Samples - most common test - use this one to compare two groups.
Paired T-test - is when you have repeated measures on a single sample, e.g. a pre- and post test score and
want to test “gain scores” of individuals vs means of two tests.
Note: Two sample T tests generally use different formula depending on whether or not the variances in the
two samples are assumed to be the same or different. An F-test is performed to choose between a “pooled
sample” variance estimate or “separate” variances. In most cases it doesn’t matter, but procedure is to use
the pooled variance estimator if the F-test of differences in variances is not significant.
One Way ANOVA: Generalization of T-test to more than two groups. Independent variable is nominal to
form groups, means are computed for each group. The F-test tests null hypothesis that all the group means
are the same. Can run “Contrasts” to do t-tests between pairs of groups and can adjust for multiple tests using
Bonferroni and related adjustments. Read about this in a statistics references if you wish.
Interval by Interval
Pearson Correlation: Run CORRELATION procedure to get the correlation coefficient between the two
variables AND a test of null hypothesis that the correlation in population is zero. Be sure you understand
distinction here between the measure of association between the two variables in the sample (correlation
coefficient) and the test of hypothesis that correlation is zero (making inference to the population).
Regression : is multivariate extension of correlation. A linear relationship between a dependent variable and
several independent variables is estimated. t-statistics for each regression coefficient test for a relationship
between X and Y while controlling for the other independent variables. Standardized regression coefficients
(betas) indicate relative importance of each independent variable. The R square statistic (use adjusted R
square) measures amount of variation in Y explained by the X’s.
Nonparametric Tests (or Distribution free statistics)
All of the above except Chi square are called parametric statistics meaning they assume interval scale
measurements and that variables are normally distributed in the population. A number of non-parametric
tests have been developed for situations where variables are measured at nominal or ordinal scales. These
tests do not assume interval scale properties. The ordinal tests are based only on rankings or “order
statistics”. The most common nonparametric tests are:
Chi square : Nominal x nominal described above.
Rank order or Spearman correlation : for two ordinal variables, Kendalls Tau if many “ties”
Mann Whitney U : corresponds to T-Test, difference in means when dependent variable is ordinal difference in ranks.
Wilcoxon matched pairs : corresponds to paired t-test with ordinal data
Kruskal Wallis one way ANOVA with ordinal data
Friedman two way ANOVA with two or more related samples and ordinal data
Others: Kolmogorov-Smirnov, Runs, Sign test
Measures of Association
These measure strength of a relationship in a sample. Most measures of association are like the correlation
coefficient. They often vary between 0 and 1 or (-1 and 1 if directional). A correlation of zero means no
(linear) association, one a perfect relationship. The preferred measures of association have a PRE
(proportionate reduction in error) interpretation. This means they tell us how much better we can predict the
dependent variable if we know the value of independent variable. See Babbie pp 416-420 or SPSS Data
Analysis Guide Chapter 19 for details.
Nominal Variables
There are several measures of association based on the Chi square statistic. Beware of Chi square as a
measure of association vs test of hypothesis as the magnitude of Chi square depends on sample size N and
number of cells in the table. You can’t compare Chi square’s across samples or tables of different size to
indicate weaker or stronger relationships.
Phi : A PRE measure for 2 by 2 tables, normalizes Chi square for sample size.The Contingency coeficient,
and Cramer’s V are not PRE measures.
Lambda is the most common PRE measure of association for nominal x nominal - based on improving
predictions of one variable knowing value of the other. Symmetric and asymmetric versions depending on
which is dependent variable.
Ordinal Variables
Gamma (Goodman and Kruskal’s gamma) - from concordant & discordant pairs
Spearman rank order correlation
Kendall’s tau-b - adjusts for ties, Tau-c, Somer’s D are NOT PRE measures.
Interval
Pearson Correlation assumes two inteval scale variables, the correlation squared gives variation
explained.
Eta for nominal or ordinal x interval, eta squared is variation in dependent variable explained by
indepedent variable.
The Normal Distribution
Important because:
1. Many continuous phenomona follow this distribution.
2. It can be used to approximate many discrete distributions.
3. Central limit theorem makes it the centerpiece of classical statistical inference. i.e., sampling distributions
are normal.
Properties: Bell shaped, mean=median=mode give center of distribution, symmetric about the mean, has
infinite range, interquartile range =1.33 std deviations, ie. plus or minus two-thirds of std. dev. from mean.
68% of values within one standard deviation of mean, 95% within two, and roughly 99% within three
standard deviations.
Probability distribution for N(,) expressed as:
Areas Under Normal Distribution
Standardizing the Normal Distribution: Standardized normal distribution has mean=0 and standard
deviation= 1-- denoted N(0,1). Can transform a random variable X with normal distribution with mean 
and standard deviation  to random variable Z with N(0,1) by the transformation, Z=(X-)/. For each X,
corresponding Z is known as its Z-score. The area under the standard normal distribution are tabulated in
tables of the standardized normal distribution. (Be careful to check exactly which areas are given.)
Sampling distribution of the mean = the distribution of means in all possible samples of size n from a
given population. There is a different distribution for each value of n, which summarizes the range of
sample means one may obtain. From this distribution we estimate the probability of obtaining a given
sample mean, i.e., determine confidence intervals for sample estimates.
Standard error of the mean is defined to be the standard deviation of the sampling distribution of the
mean.
Central limit theorem. When drawing samples of size n from a population with mean  and standard
deviation  , the sampling distribution of the mean approaches a normal distribution with mean equal to the
population mean  and standard deviation equal to [/n]. Hence, the standard error of the mean is
[/n], where  is the standard deviation in the population and n is the sample size.
Sampling from finite populations: The above formula assumes sampling with replacement. Most surveys
don't sample with replacement. For small populations (N), one must correct for difference by the finite
population correction factor(fpc) , fpc = sqrt[(N-n)/(N-1)]. Standard error of mean becomes
[/n] *fpc.
Sample size determination for the mean: To estimate the sample size needed to achieve a given level of
accuracy and confidence in the estimate of the mean, you need
1. Confidence level desired- this determines the value of Z.
2. Sampling error permitted, e.
3. Standard deviation in the population, .
Then, necessary sample size is: n=Z²²/e².
Transform this equation to estimate sampling error for a given sample size, population variance &
confidence level.
e = (Z * )/ n] ,
For Z=2 (95% confidence level), e= 2 * )/ n] , two standard errors.
11. SPSS procedures: Basics
a. Descriptive statistics
FREQUENCIES for nominal/ordinal (limited number of values). /STATISTICS SEMEAN to construct
confidence intervals.
DESCRIPTIVES for interval scale. /STATISTICS SEMEAN for standard error of mean.
b. Hypothesis testing : two variable relationships CROSSTABS : Table displays of two or more variables
measured at nominal or ordinal level. Produces a variety of statistics including Chi square, phi, lambda,
gamma, etc with/STATISTICS ALL.
T-TEST tests for difference in mean between two groups either independent samples or paired
t-test.
ONEWAY. ONEWAY compares means across more than two groups and does pairwise T-Tests
(CONTRASTS). Can also compare means and variances for population
subgroups with MEANS.
c. Correlation
CORRELATION for interval scale variables. Gives pearson correlation coefficient and a
test of whether it is significantly different from zero.
PLOT two variable printer plots
d. NPAR TESTS a collection of nonparametric tests including Kolmogorov-Smirnov, McNemar, Sign,
Wilcoxon, Cochrane, Friedman, Mann-Whitney, Wald-Wolfowitz, and others.
d. REGRESSION for estimating linear models and testing a variety of hypotheses
DISCRIMINANT analysis for linear models to predict group membership
e. ANOVA for general analysis of variance, MANOVA multivariate
f. FACTOR ANALYSIS : multivariate method for identifying a set of underlying dimensions of a data set
or reducing many variables to a smaller number of orthogonal dimensions