Final review - statistics Spring 03

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Final review - statistics Spring 03

• Also, see final review - research design

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Statistics
Descriptive Statistics
Statistics to summarize and describe the data we
collected
Inferential Statistics
Statistics to make inferences from samples to the
populations
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Descriptive Statistics
A summary of your data
Center / Central Tendencies Indicates a central
value for the variable Measures of Dispersion
(Variability / Spread) Indicate how much each
participants score vary from each other Measures
of Association Indicates how much variables go
together (Shown in Tables, Graphs,
Distributions)
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Measures of Center

• Mode
• A value with the highest frequency
• The most common value
• Median
• The middle score
• Mean
• Average

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WHY are LEVELS / SCALE of MEASUREMENT IMPORTANT?

• Because you need to match the statistic you use
  to the kind of variable you have

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Measures of Central Tendency, Center
Nominal
Ordinal
Interval/Ratio
Mean
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Summary
Meaningful Zero
Equal Interval
Ratio
Interval
Info of difference among values
Order
Ordinal
Difference
Nominal
Level of Measurement
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Why Equal Distance Matters?

• If the distance between values are equal (as in
  interval or ratio data), you are able to
  calculate (add, subtract, multiply, divide) values

• You can get a mean only for interval/ratio
  variables
• A wider variety of statistical tests are
  available for interval/ratio variables

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4 5 6 7 8 9 10
What are the Mean, Median, and Mode for this
distribution?
What is this distribution shape called?
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Types of Measures of Dispersion Variability /
Spread

• Frequencies / Percentages
• Range
• The distance between the highest score and the
  lowest score (highest lowest)
• Standard deviation /
• Variance

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Variance / Standard Deviation

• Variance (S-squared) An approximate average of
  the squared deviations from the mean
• Standard Deviation(S or SD) Square root of
  variance
• The larger the variance/ SD is, the higher
  variability the data has or larger variation in
  scores, or distributions that vary widely from
  the mean.

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Measures of Dispersion
Nominal
Ordinal
Interval/Ratio
StandardDeviatn, Variance
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CORRELATION

• Co-relation
• 2 variables tend to go together
• Indicates how strongly and
• in which direction two variables are correlated
  with each other
• Correlation does NOT EQUAL cause

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SIGN

• 0 No systematic relationship

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Correlation Co-efficient
Negative
Positive
1
-1
0
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SIZE

• Ranges from 1 to 1
• 0 or close to 0 indicates NO relationship
• /- .2 - .4 weak
• /- .4 - .6 moderate
• /- .6 - .8 strong
• /- .8 - .9 very strong
• /- 1.00 perfect
• Negative relationships are NOT weaker!

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Significance Test

• Correlation co-efficient also comes with
  significance test (p-value)
• p.05 .05 probability of no correlation in the
  population 5 risk of TYPE I Error 95
  confidence level
• If plt.05, reject H0 and support Ha at 95
  confidence level

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Inferential Statistics

1. Infer characteristics of a population from the
   characteristics of the samples.
2. Hypothesis Testing
3. Statistical Significance
4. The Decision Matrix

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Inferential Statistics
Sample Statistics X SD n
Population Parameters m s N
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Inferential Statistics

• assess -- are the sample statistics indicators
  of the population parameters?
• Differences between 2 groups -- happened by
  chance?
• What effect do random sampling errors have on our
  results?

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Random sampling error

• ?Random sampling error
• Difference between the sample characteristics
  and the population characteristics caused by
  chance

• Sampling bias
• Difference between the sample characteristics
  and the population characteristics
• caused by biased (non-random) sampling

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Probability

• Probability (p) ranges between 1 and 0
• p 1 means that the event would occur in every
  trial
• p 0 means the event would never occur in any
  trial
• The closer the probability is to 1, the more
  likely that the event will occur
• The closer the probability is to 0, the less
  likely the event will occur

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P gt .05 means that

• Means of two groups fall in 95 central area of
  normal distribution with one population mean

Mean 1
Mean 2
95
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P lt .05 means that

• Means of two groups do NOT fall in 95 central
  area of normal distribution of one population
  mean, so it is more reasonable to assume that
  they belong to different populations

?1
?2
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Null Hypothesis

• Says IV has no influence on DV
• There is no difference between the two variables.
• There is no relationship between the two
  variables.

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Null Hypothesis

• States there is NO true difference between the
  groups
• If sample statistics show any difference, it is
  due to random sampling error
• Referred as H0
• (Research Hypothesis Ha)
• If you can reject H0, you can support Ha
• If you fail to reject H0, you reject Ha

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• Be conservative.
• What are chances I would get these results if
  null hypothesis is true?
• Only if pattern is highly unlikely (p ? .05) do
  you reject null hypothesis and support your
  hypothesis
• Since cannot be 100 sure your conclusion is
  correct, you take up to 5 risk.
• Your p-value tells you the risk /the
• probability of making TYPE I Error

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True state
Wrong person to marry
Right person to marry
Type II error
You think its the wrong person to marry
Type I error
You think - right person to marry
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True state
Fire
No fire
Type II error
No Alarm
Type I error
Alarm
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True State
Fire
Ho (no fire)
Ha
You decide...
Accept Ho (no alarm)
Type II error
Type I error
Reject Ho
Ho null hypothesis there is NO fire
Alarm
Ha alternative hyp. there IS a FIRE
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Easy ways to LOSE points

• Use the word prove
• Better to say support the hypothesis or
• consistent with the hypothesis
• Tentative statements acknowledge possibility of
  making a Type 1 or Type 2 error
• Use the word random incorrectly

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Significance Test

• Significance test examines the probability of
  TYPE I error (falsely rejecting H0)
• Significance test examines how probable it is
  that the observed difference is caused by random
  sampling error
• Reject the null hypothesis if probability is lt.05
  (probability of TYPE I error
• is smaller than .05)

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Principle Logic
P lt .05
Reject Null Hypothesis (H0)
Support Your Hypothesis (Ha)
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Logic of Hypothesis Testing
Statistical tests used in hypothesis testing deal
with the probability of a particular event
occurring by chance.
Is the result common or a rare occurrence if
only chance is operating?
A score (or result of a statistical test) is
Significant if score is unlikely to occur on
basis of chance alone.
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Level of Significance
The Level of Significance is a cutoff point for
determining significantly rare or unusual scores.
Scores outside the middle 95 of a distribution
are considered Rare when we adopt the standard
5 Level of Significance This level of
significance can be written as p .05
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Decision Rules

• Reject Ho (accept Ha) when
• the sample statistic is statistically significant
  at the
• chosen p level, otherwise accept Ho (reject Ha).
  
• Possible errors
• You reject the Null Hypothesis when in fact it is
  true,
• a Type I Error, or Error of Rashness.
• You accept the Null Hypothesis when in fact it is
  false,
• a Type II Error, or Error of Caution.

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True state
Your decision
?
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Parametric Tests
To compare two groups on Mean Scores use t-test.
For more than 2 groups use Analysis of Variance
(ANOVA)
Nonparametric Tests
Cant get a mean from nominal or ordinal data.
Chi Square tests the difference in Frequency
Distributions of two or more groups.
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Parametric Tests

• Used with data w/ mean score or standard
  deviation.
• t-test, ANOVA and Pearsons Correlation r.
• Use a t-test to compare mean differences between
  two groups (e.g., male/female and
  married/single).

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Parametric Tests

• use ANalysis Of VAriance (ANOVA) to compare more
  than two groups (such as age and family income)
  to get probability scores for the overall group
  differences.
• Use a Post Hoc Tests to identify which subgroups
  differ significantly from each other.

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When comparing two groups on MEAN SCORES use the
t-test.
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T-test

• If plt.05, we conclude that two groups are drawn
  from populations with different distribution
  (reject H0) at 95 confidence level

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Our Research Hypothesis hair length leads to
different perceptions of a person. The Null
Hypothesis there will be no difference between
the pictures.
When comparing two groups on MEAN SCORES use the
t-test.
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I think she is one of those people who quickly
earns respect.
Short Hair Mean 2.2 SD
1.9 n 100
p .03
Accept Ha
Mean scores come from different distributions.
Long Hair Mean 4.1 SD
1.8 n 100
Accept Ho
Mean scores reflect just chance differences from
a single distribution.
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In my opinion, she is a mature person.
Short Hair Mean 1.6 SD
1.7 n 100
p .01
Accept Ha
Mean scores come from different distributions.
Long Hair Mean 3.6 SD
1.2 n 100
Accept Ho
Mean scores reflect just chance differences from
a single distribution.
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I think we are quite similar to one another.
Short Hair Mean 3.7 SD
1.8 n 100
Accept Ha
Mean scores come from different distributions.
p .89
Long Hair Mean 3.9 SD
1.5 n 100
Accept Ho
Mean scores are just chance differences from a
single distribution.
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A nonsignificant result may be caused by a

• A. low sample size.
• B. very cautious significance level.
• C. weak manipulation of independent variables.
• D. true null hypothesis.

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When to use various statistics

• Parametric
• Interval or ratio data

• Non-parametric
• Ordinal and nominal data

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Chi-Square X2

• Chi Square tests the difference in frequency
  distributions of two or more groups.
• Test of Significance
• of two nominal variables or
• of a nominal variable an ordinal variable
• Used with a cross tabulation table