statistics primer orc staff: xin xin (cindy) ryan glaman brett kellerstedt 1
TRANSCRIPT
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Statistics Primer
ORC Staff:
Xin Xin (Cindy)
Ryan Glaman
Brett Kellerstedt
Quick Overview of Statistics
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Descriptive vs. Inferential Statistics
Descriptive Statistics: summarize and describe data (central tendency, variability, skewness)
Inferential Statistics: procedure for making inferences about population parameters using sample statistics
Sample
Population
Measures of Central Tendency
Raw data Simple frequency distribution
Group frequency distribution
Notations
Mode Pick out the value (s) occurring more than any other value.
Pick out the value (s) with the highest frequency.
= Difference between the freq. of modal class and the freq. of the next lower class.
= Difference between the freq. of modal class and the freq. of the next higher class.
L1 = Lower class boundary of the modal class
c = class width of the modal class
Median
1.Order data2.Determine
median position = (n+1)/2
3.Locate median based on step 2.
1.Order data2.Determine median
position = (n+1)/23.Locate median
based on step 2 using the freq. column
Lm=lower class boundary of median classn = sample sizeC.F. = sum of all frequencies lower than the median classfmed = frequency of the median class
c = class width of the median class
Mean Add up all the data values and divide by the number of values.
Find the product of all the values and their frequencies ; then add all the products; and finally divide by the total frequency.
Find the product of all the midpoints and their frequencies ; then add all the products; and finally divide by the total frequency.
X = the actual values (for raw data and ungrouped freq. dist.)
= midpoints (for group freq. dist.)f = frequency n = sample sizeN = population size
= summation or sum of
c
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11L
4
1
2
cf
FCnL
medm
.).2(
n
Xor
N
X
f
fX
f
fX
Description Applicability Advantage Disadvantage
Range Difference between the largest and the smallest value in the data.
1.Interval/ratio2.No outliers exist
1.Simple to calculate
1.Highly influenced by outliers.
2.Does not use all data
Mean deviation
It measures the average absolute deviations from the mean. Uncommonly used
1.Interval/ratio2.When no outliers
exist
1.Use all the data2.Easy to interpret
1.Not resistant to outliers
2.Does not yield any further useful statistical properties.
Variance/ standard deviation
Variance is the average squared deviations from the mean.
Standard deviation is square root of the variance. Commonly used.
1.Interval/ratio2.When no outliers
exist
1.Provides good statistical properties, by avoiding the use of absolute values.
2.Use all the data
1.Not resistant to outliers.
2.Variance depends on the units of measurement, therefore not easy to make comparisons.
Sum of Squares
Measures variability of the scores, the total variation of all scores
1.Interval/ratio2.When no outliers
exist
1.Effect size calculation
1.Not resistant to outliers.
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Measures of Variability5
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Variance and Sum of Squares
2xxSS
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2
2
n
xxS
1
2
n
xxS
x xx 2xx 6 1 1 5 0 0 3 -2 4 5 0 0 6 1 1
Mean = 5
Empirical Rule
The empirical rule states that symmetric or normal distribution with population mean μ and standard deviation σ have the following properties.
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Outcome Ball 1 Ball 2 Mean
1 1 1 1.0
2 1 2 1.5
3 1 3 2.0
4 2 1 1.5
5 2 2 2.0
6 2 3 2.5
7 3 1 2.0
8 3 2 2.5
9 3 3 3.0
All possible outcomes are shown below in Table 1.
Table 1. All possible outcomes when two balls are sampled with replacement.
Sampling Distribution
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Sampling Error
As has been stated before, inferential statistics involve using a representative sample to make judgments about a population. Lets say that we wanted to determine the nature of the relationship between county and achievement scores among Texas students. We could select a representative sample of say 10,000 students to conduct our study. If we find that there is a statistically significant relationship in the sample we could then generalize this to the entire population.
However, even the most representative sample is not going to be exactly the same as its population. Given this, there is always a chance that the things we find in a sample are anomalies and do not occur in the population that the sample represents. This error is referred as sampling error.
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Sampling Error
A formal definition of sampling error is as follows:Sampling error occurs when random chance produces a sample statistic that is not equal to the population parameter it represents.
Due to sampling error there is always a chance that we are making a mistake when rejecting or failing to reject our null hypothesis.
Remember that inferential procedures are used to determine which of the statistical hypotheses is true. This is done by rejecting or failing to reject the null hypothesis at the end of a procedure.
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Sampling Distribution and Standard Error (SE)
https://www.youtube.com/watch?v=hvIDuEmWt2k
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Hypothesis Testing
Null Hypothesis Statistical Significance Testing (NHSST)
Testing p-values using statistical significance tests
Effect Size
Measure magnitude of the effect (e.g., Cohen’s d)
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Null Hypothesis Statistical Significance Testing
Statistical significance testing answers the following question:
Assuming the sample data came from a population in which the null hypothesis is exactly true, what is the probability of obtaining the sample statistic one got for one’s sample data with the given sample size? (Thompson, 1994)
Alternatively:
Statistical significance testing is used to examine a statement about a relationship between two variables.
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Hypothetical Example
Is there a difference between the reading abilities of boys and girls?
Null Hypothesis (H0): There is not a difference between the reading abilities of boys and girls.
Alternative Hypothesis (H1): There is a difference between the reading abilities of boys and girls.
Alternative hypotheses may be non-directional (above) or directional (e.g., boys have a higher reading ability than girls).
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Testing the Hypothesis
Use a sampling distribution to calculate the probability of a statistical outcome.
pcalc = likelihood of the sample’s result
pcalc < pcritical: reject H0
pcalc ≥ pcritical: fail to reject H0
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Level of Significance (pcrit)
Alpha level (α) determines:
The probability at which you reject the null hypothesis
The probability of making a Type I error (typically .05 or .01)
True Outcome in Population
Reject H0 is true
H0 is false
Observed Outcome
Reject H0 Type I error (α) Correct Decision
Fail to reject H0
Correct Decision
Type II error (β)
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Example: Independent t-test
Research Question: Is there a difference between the reading abilities of boys and girls?
Hypotheses:
H0: There is not a difference between the reading abilities of boys and girls.
H1: There is a difference between the reading abilities of boys and girls.
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Dataset
Reading test scores (out of 100)
Boys Girls
88 88
82 90
70 95
92 81
80 93
71 86
73 79
80 93
85 89
86 87
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Significance Level
α = .05, two-tailed test
df = n1 + n2 – 2
= 10 + 10 – 2 = 18
Use t-table to determine tcrit
tcrit = ±2.101
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Decision Rules
If tcalc > tcrit, then pcalc < pcrit
Reject H0
If tcalc ≤ tcrit, then pcalc ≥ pcrit
Fail to reject H0
-2.101 2.101
p = .025 p = .025
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Computations
Boys Girls
Frequency (N) 10 10
Sum (Σ) 807 881
Mean () 80.70 88.10
Variance (S2) 55.34 26.54
Standard Deviation (S) 7.44 5.15
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Computations cont.
Pooled variance
Standard Error
= 40.944
= 2.862
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Computations cont.
Compute tcalc
Decision: Reject H0. Girls scored statistically significantly higher on the reading test than boys did.
= -2.586𝑡=𝑋 1−𝑋 2
𝑆𝐸 𝑋1− 𝑋2
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Confidence Intervals
Sample means provide a point estimate of our population means. Due to sampling error, our sample estimates may not perfectly represent our populations of interest. It would be useful to have an interval estimate of our population means so we know a plausible range of values that our population means may fall within.
95% confidence intervals do this.
Can help reinforce the results of the significance test.
CI95 = ± tcrit (SE)
= -7.4 ± 2.101(2.862) = [-13.412, -1.387]
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Statistical Significance vs. Importance of Effect
Does finding that p < .05 mean the finding is relevant to the real world?
Not necessarily…
https://www.youtube.com/watch?v=5OL1RqHrZQ8
Effect size provides a measure of the magnitude of an effect
Practical significance
Cohen’s d, η2, and R2 are all types of effect sizes
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Cohen’s d
Equation:
Guidelines: d = .2 = small
d = .5 = moderate
d = .8 = large
Not only is our effect statistically significant, but the effect size is large.
= -1.16