Descriptive Statistics

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Descriptive Statistics
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Five types of statistical analysis
What are the characteristics of the respondents?
Descriptive
What are the characteristics of the population?
Inferential
Are two or more groups the same or different?
Differences
Are two or more variables related in a systematic
way?
Associative
Can we predict one variable if we know one or
more other variables?
Predictive
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Descriptive Statistics

• Summarization of a collection of data in a clear
  and understandable way
• the most basic form of statistics
• lays the foundation for all statistical knowledge

• Measures of central tendency
• mean, median, mode
• Measures of dispersion
• range, standard deviation, and coefficient of
  variation
• Measures of shape
• skewness and kurtosis

• If you use fewer statistics to describe the
  distribution of a variable, you lose information
  but gain clarity.

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Type of Measurement
Type of descriptive analysis
Frequency table Proportion (percentage) Frequenc
y table Category proportions (percentages) Mode
Two categories
Nominal
More than two categories
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Type of Measurement
Type of descriptive analysis
Ordinal
Rank order Median
Interval
Arithmetic mean
Ratio
means
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Data Tabulation

• Tabulation The organized arrangement of data in
  a table format that is easy to read and
  understand.
• A count of the number of responses to each
  question.
• Simple Tabulation tabulating of results of only
  one variable informs you how often each response
  was given.
• Frequency Distribution A distribution of data
  that summarizes the number of times a certain
  value of a variable occurs expressed in terms of
  percentages.

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Frequency Tables

• The arrangement of statistical data in a
  row-and-column format that exhibits the count of
  responses or observations for each category
  assigned to a variable
• How many of certain brand users can be called
  loyal?
• What percentage of the market are heavy users and
  light users?
• How many consumers are aware of a new product?
• What brand is the Top of Mind of the market?

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More on relative frequency distributions

• Rules for relative frequency distributions
• Make sure each observation is in one and only one
  category.
• Use categories of equal width.
• Choose an appealing number of categories.
• Provide labels
• Double-check your graph.

A bar graph is a relative frequency distribution
of a qualitative variable
A histogram is a relative frequency distribution
of a quantitative variable
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How many times per week do you use mouthwash
? 1__ 2__ 3__ 4__ 5__ 6__ 7__ 1 1 2 2 2 3 3
3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 6 6 6 7 7
1 2
2 3
3 5
4 7
5 5
6 3
7 2
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Normal Distribution
?
- ?
?
?
a
b
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The total area under the curve is equal to 1,
i.e. It takes in all observations The area of a
region under the normal distribution between any
two values equals the probability of observing a
value in that range when an observation is
randomly selected from the distribution For
example, on a single draw there is a 34 chance
of selecting from the distribution a person with
an IQ between 100 and 115
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• Curve is basically bell shaped from - ? to ?
• symmetric with scores concentrated in the middle
  (i.e. on the mean) than in the tails.
• Mean, medium and mode coincide
• They differ in how spread out they are.
• The area under each curve is 1.
• The height of a normal distribution can be
  specified mathematically in terms of two
  parameters the mean (m) and the standard
  deviation (s).

Normal Distributions
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Skewed Distributions

• Occur when one tail of the distribution is longer
  than the other.
• Positive Skew Distributions
• have a long tail in the positive direction.
• sometimes called "skewed to the right"
• more common than distributions with negative
  skews
• E.g. distribution of income. Most people make
  under 80,000 a year, but some make quite a bit
  more with a small number making many millions of
  dollars per year
• The positive tail therefore extends out quite a
  long way
• Negative Skew Distributions
• have a long tail in the negative direction.
• called "skewed to the left."
• negative tail stops at zero
• E.g. GPA

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• Kurtosis how peaked a distribution is. A zero
  indicates normal distribution, positive numbers
  indicate a peak, negative numbers indicate a
  flatter distribution)

Peaked distribution
Flat distribution
Thanks, Scott!
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Summary statistics

• central tendency
• Dispersion or variability
• A quantitative measure of the degree to which
  scores in a distribution are spread out or are
  clustered together

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Measures of Central Tendency

• Mode the number that occurs most often in a
  string (nominal data)
• Median half of the responses fall above this
  point, half fall below this point (ordinal data)
• Mean the average (interval/ratio data)

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• Mode
• the most frequent category
• users 25
• non-users 75
• Advantages
• meaning is obvious
• the only measure of central tendency that can be
  used with nominal data.
• Disadvantages
• many distributions have more than one mode, i.e.
  are multimodal
• greatly subject to sample fluctuations
• therefore not recommended to be used as the only
  measure of central tendency.

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• Median
• the middle observation of the data
• number times per week consumers use mouthwash
• 1 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 6 6 6
  7 7

Frequency distribution of Mouthwash use per week
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Measures of Dispersion or Variability

• Minimum, Maximum, and Range (Highest value minus
  the lowest value)
• Variance
• Standard Deviation (A measures distance from the
  mean)

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- 1 SD
1 SD
RANGE
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Variance

• The difference between an observed value and the
  mean is called the deviation from the mean
• The variance is the mean squared deviation from
  the mean
• i.e. you subtract each value from the mean,
  square each result and then take the average.
• Because it is squared it can never be negative

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Standard Deviation

• The standard deviation is the square root of the
  variance
• Thus the standard deviation is expressed in the
  same units as the variables
• Helps us to understand how clustered or spread
  the distribution is around the mean value.

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Normal Distributions with different SD
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Cross Tabulation

• A statistical technique that involves tabulating
  the results of two or more variables
  simultaneously
• informs you how often each response was given
• Shows relationships among and between variables
• frequency distribution for each subgroup compared
  to the frequency distribution for the total
  sample
• must be nominally scaled

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Cross-tabulation

• Helps answer questions about whether two or more
  variables of interest are linked
• Is the type of mouthwash user (heavy or light)
  related to gender?
• Is the preference for a certain flavor (cherry or
  lemon) related to the geographic region (north,
  south, east, west)?
• Is income level associated with gender?
• Cross-tabulation determines association not
  causality.

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Dependent and Independent Variables

• The variable being studied is called the
  dependent variable or response variable.
• A variable that influences the dependent variable
  is called independent variable.

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Cross-tabulation

• Cross-tabulation of two or more variables is
  possible if the variables are discrete
• The frequency of one variable is subdivided by
  the other variable categories.
• Generally a cross-tabulation table has
• Row percentages
• Column percentages
• Total percentages
• Which one is better?
• DEPENDS on which variable is considered as
  independent.

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Contingency Table

• A contingency table shows the conjoint
  distribution of two discrete variables
• This distribution represents the probability of
  observing a case in each cell
• Probability is calculated as

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Cross tabulation
GROUPINC Gender Crosstabulation
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General Procedure for Hypothesis Test

• Formulate H0 (null hypothesis) and H1
  (alternative hypothesis)
• Select appropriate test
• Choose level of significance
• Calculate the test statistic (SPSS)
• Determine the probability associated with the
  statistic.
• Determine the critical value of the test
  statistic.

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General Procedure for Hypothesis Test

• a) Compare with the level of significance, ?
• b) Determine if the critical value falls in
  the rejection region. (check tables)
• Reject or do not reject H0
• Draw a conclusion

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1. Formulate H1and H0

• The hypothesis the researcher wants to test is
  called the alternative hypothesis H1.
• The opposite of the alternative hypothesis is the
  null hypothesis H0 (the status quo)(no difference
  between the sample and the population, or between
  samples).
• The objective is to DISPROVE the null hypothesis.
  
• The Significance Level is the Critical
  probability of choosing between the null
  hypothesis and the alternative hypothesis

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2. Select Appropriate Test

• The selection of a proper Test depends on
• Scale of the data
• nominal
• interval
• the statistic you seek to compare
• Proportions (percentages)
• means
• the sampling distribution of such statistic
• Normal Distribution
• T Distribution
• ?2 Distribution
• Number of variables
• Univariate
• Bivariate
• Multivariate
• Type of question to be answered

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Example A tire manufacturer believes that men are
more aware of their brand than women. To find
out, a survey is conducted of 100 customers, 65
of whom are men and 35 of whom are women. The
question they are asked is Are you aware of our
brand Yes or No. 50 of the men were aware and
15 were not, whereas 10 of the women were aware
and 25 were not. Are these differences
significant?
Men Women Total
Aware 50 10 60 Unaware
15 25 40 65 35
100
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1. Formulate H1and H0

• We want to know whether brand awareness is
  associated with gender. What are the Hypotheses

H0 H1
There is no difference in brand awareness based
on gender There is a difference in brand
awareness based on gender
Chi-square test results are unstable if cell
count is lower than 5
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2. Select Appropriate Test
X2 (Chi Square)

• Used to discover whether 2 or more groups of one
  variable (dependent variable) vary significantly
  from each other with respect to some other
  variable (independent variable).
• Are the two variables of interest associated
• Do men and women differ with respect to product
  usage (heavy, medium, or light)
• Is the preference for a certain flavor (cherry or
  lemon) related to the geographic region (north,
  south, east, west)?
• H0 Two variables are independent (not
  associated)
• H1 Two variables are not independent
  (associated)
• Must be nominal level, or, if interval or ratio
  must be divided into categories

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Awareness of Tire Manufacturers Brand
Men Women Total
Aware 50/39 10/21 60 Unaware
15/26 25/14 40 65
35 100
Estimated cell Frequency
Ri total observed frequency in the ith row Cj
total observed frequency in the jth column n
sample size Eij estimated cell frequency
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3. Choose Level of Significance

• Whenever we draw inferences about a population,
  there is a risk that an incorrect conclusion will
  be reached
• The real question is how strong the evidence in
  favor of the alternative hypothesis must be to
  reject the null hypothesis.
• The significance level states the probability of
  rejecting H0 when in fact it is true.
• In this example an error would be committed if we
  said that there is a difference between men and
  women with respect to brand awareness when in
  fact there was no difference i.e. we have
  rejected the null hypothesis when it is in fact
  true
• This error is commonly known as Type I error, The
  value of ? is called the significance level of
  the test Type I error

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• Significance Level selected is typically .05 or
  .01
• i.e 5 or 1
• In other words we are willing to accept the risk
  that 5 (or 1) of the time the results we get
  indicate that we should reject the null
  hypothesis when it is in fact true.
• 5 (or 1) of the time we are willing to commit
  a Type 1 error
• stating there is a difference between men and
  women with respect to brand awareness when in
  fact there is no difference

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3. Choose Level of Significance

• We commit Type error II when we incorrectly
  accept a null hypothesis when it is false. The
  probability of committing Type error II is
  denoted by ?.
• In our example we commit a type II error when we
  say that.

there is NO difference between men and women with
respect to brand awareness (we accept the null
hypothesis) when in fact there is
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Type I and Type II Errors
Accept null
Reject null
Null is true
Correct- no error
Type I error
Null is false
Type II error
Correct- no error
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Which is worse?

• Both are serious, but traditionally Type I error
  has been considered more serious, thats why the
  objective of hypothesis testing is to reject H0
  only when there is enough evidence that supports
  it.
• Therefore, we choose ? to be as small as possible
  without compromising ?. (accepting when false)
• Increasing the sample size for a given a will
  decrease ß (I.e. accepting the null hypothesis
  when it is in fact false)

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Awareness of Tire Manufacturers Brand
Men Women Total
Aware 50/39 10/21 60 Unaware
15/26 25/14 40 65
35 100
Estimated cell Frequency
Ri total observed frequency in the ith row Cj
total observed frequency in the jth column n
sample size Eij estimated cell frequency
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Chi-Square Test
Estimated cell Frequency
Ri total observed frequency in the ith row Cj
total observed frequency in the jth column n
sample size Eij estimated cell frequency
Chi-Square statistic
x² chi-square statistics Oi observed
frequency in the ith cell Ei expected frequency
on the ith cell
Degrees of Freedom
d.f.(R-1)(C-1)
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• Degrees of Freedom
• the number of values in the final calculation of
  a statistic that are free to vary
• For example To calculate the standard deviation
  of a random sample, we must first calculate the
  mean of that sample and then compute the sum of
  the squared deviations from that mean

• While there will be n such squared deviations
  only (n - 1) of them are free to assume any value
  whatsoever.
• This is because the final squared deviation from
  the mean must include the one value of X such
  that the sum of all the Xs divided by n will
  equal the obtained mean of the sample.
• All of the other (n - 1) squared deviations from
  the mean can, theoretically, have any values
  whatsoever..

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4. Calculate the Test Statistic
Chi-Square Test Differences Among Groups
Chi-square test results are unstable if cell
count is lower than 5
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5. Determine the Probability-value (Critical
Value)

• The p-value is the probability of seeing a random
  sample at least as extreme as the sample observed
  given that the null hypothesis is true.
• given the value of alpha, ? we use statistical
  theory to determine the rejection region.
• If the sample falls into this region we reject
  the null hypothesis otherwise, we accept it
• Sample evidence that falls into the rejection
  region is called statistically significant at the
  alpha level.

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COMBINATIONS
A combination is the selection of a certain
number of objects taken from a group of objects
without regard to order. We use the symbol
(5 choose 3) to indicate that we have five
objects taken three at a time, without regard to
order. To calculate the possible number of
combinations the formula used is
5x4x3x2x1 120 10
(3x2x1)x(2x1) 12 If we choose a sample
of 5 from a total of 20 there are 15, 504
possible combinations. If we took the means of
some measurement for each of the possible
combinations those means would form a normal
distribution.
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Critical value

• A critical value is the value that a test
  statistic must exceed in order for the the null
  hypothesis to be rejected.
• For example, the critical value of t (with 12
  degrees of freedom using the .05 significance
  level) is 2.18.
• This means that for the probability value to be
  less than or equal to .05, the absolute value of
  the t statistic must be 2.18 or greater.

critical value
Significance level (.05)
Test statistic
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Significance from p-values -- continued

• How small is a small p-value? This is largely a
  matter of semantics but if the
• p-value is less than 0.01, it provides
  convincing evidence that the alternative
  hypothesis is true
• p-value is between 0.01 and 0.05, there is
  strong evidence in favor of the alternative
  hypothesis
• p-value is between 0.05 and 0.10, it is in a
  gray area
• p-values greater than 0.10 are interpreted as
  weak or no evidence in support of the
  alternative.

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5. Determine the Probability-value (Critical
Value)
Chi-square Test for Independence
Under H0, the probability distribution is
approximately distributed by the Chi-square
distribution (?2).
Chi-square
?2 with 1 d.f. at .05 critical value 3.84
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• a) Compare with the level of significance, ?
• b) Determine if the critical value falls in
  the rejection region. (check tables)
• 22.16 is greater than 3.84 and falls in the
  rejection area
• In fact it is significant at the .001 level,
  which means that the chance that our variables
  are independent, and we just happened to pick an
  outlying sample, is less than 1/1000
• Or, in other words, the chance that we have a
  Type 1 error is less than .1
• i.e. That there is a .1 chance that we reject
  the null hypothesis when it is true -- that there
  is no difference between men and women with
  respect to brand awareness, and say that there
  is, when in fact the null hypothesis is true
  there is no difference.

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• Reject or do not reject H0
• Since 22.16 is greater than 3.84 we reject the
  null hypothesis
• Draw a conclusion
• Men and women differ with respect to brand
  awareness, specifically, men are more brand aware
  then women