STATISTICS
Formulas
Standard Normal Table
t-Distribution Table
Open program on same page
Descriptive Statistics:
Constructing Relative Frequency Distribution
Video
Note: Construct Frequency Table, Relative Frequency Table, and Degrees for Pie Chart
Bar Chart or Colum Chart
Video
Note: Draw Bar Chart or Colum Chart
Line Chart or Time-Series Graph
Video
Note: Draw Line Chart or Time-Series Graph
Pie Chart
Video
Note: Draw Pie Chart
Histogram
Video
Note: Draw Histogram
Box Plot
Video
Note: Draw Boxplot
Mean, Variance, Standard Deviation, Five-number Summary, Outliers (Free-Form Version)
Video
Note: Calculate x̄, s, s
2
, Q
1
, Q
2
, Q
3
, μ, σ, and σ
2
for ungrouped data
Mean, Variance, Standard Deviation, Five-number Summary, Outliers (Excel-Grid Version)
Video
Note: Calculate x̄, s, s
2
, Q
1
, Q
2
, Q
3
, μ, σ, and σ
2
for ungrouped data
Mean, Variance, and Standard Deviation of Grouped Data
Video
Note: Calculate x̄, s, s
2
, μ, σ, and σ
2
for grouped data
Mean, Variance, and Standard Deviation of the differences of Two Sets of Data
Note: Calculating
d̄
= Mean of differences of two sets of data ;
and s
d
= standard deviation of the differences
Empirical Rule and Bell-Shaped Distribution (Computational Program)
Video
Empirical Rule and Bell-Shaped Distribution (Simulation Program)
Video
Chebyshev's Inequality (Computational Program)
Video
Chebyshev's Inequality (Simulation Program)
Video
Five-number Summary and Boxplot
Video
Note: Calculate Mininum, Q
1
, Q
2
, Q
3
, Maximum and draw boxplot.
Finding
k
th Percentile
Video
Correlation and Scatter Diagram:
Correlation Coefficient and Regression Line (new version)
Video 1
Video 2
Video 3
Correlation Coefficient and Regression Line
Video 1
Video 2
Video 3
Probability, Permutation, Combination:
Probability
Probability of Compound Events
Probability of Independent Events
Conditional Probability
Permutations and Combinations of n Distinct Objects Taken x at a Time
Video
Find
n
C
r
and
n
P
r
Discrete and Binomial Distribution:
Discrete Random Variable: Mean, Variance, and Standard Deviation
Video
Note: Program calculates mean, variance, and standard deviation
Binomial Distribution
Video
Note: Program calculates P(X = c), P(X < c), P(X > c), P(X ≤ c), P(X ≥ c), P(a ≤ X ≤ c)
Normal Distribution:
Find Left-Tailed Area Given Score
Video(1)
Video(2)
Find Right-Tailed Area Given Score
Video(1)
Video(2)
Find Area Between Two Scores
Video(1)
Video(2)
Find Score Given Left-Tailed Area
Video(1)
Video(2)
Find Score Given Right-Tailed Area
Video(1)
Video(2)
Find Scores Gven Left-Tailed Area and Right-Tailed Area
Video(1)
Video(2)
Simulation of Two Normal Distributions
Assessing Normality (new version)
Video
Assessing Normality
Video
Normal Distribution vs. Binomial Distribution:
Binomial Distribution vs. Normal Distribution
Simulation of Binomial Distribution vs. Normal Distribution
The Normal Approximation to the Binomial Probability Distribution
(new version)
Video
The Normal Approximation to the Binomial Probability Distribution
(old version)
Video
Sampling Distribution:
Sampling Distribution of the Sample Mean ( x̅ )
(computational version)
Video
Note: Given sample size (n), use normal distribution to find P(X̅ < c), P(X̅ ≤ c), P(X̅ > c), or P(X̅ ≥ c)
Sampling Distribution of the Sample Proportion ( p̂ )
(computational version)
Video
Note: Given sample size (n), use normal distribution to find P(p̂ < c), P(p̂ ≤ c), P(p̂ > c), or P(p̂ ≥ c)
Sampling Distribution of the Sample Mean ( x̅ )
(simulation version)
Video
Sampling Distribution of the Sample Proportion ( p̂ )
(simulation version)
Video
t-Distribution:
Simulation of t-Distribution vs. Standard Normal Distribution
Simulation of t-Distribution vs. Standard Normal Distribution (Animated Version)
Find Left-Tailed Area Given t-score
Video
Find Right-Tailed Area Given t-score
Video
Find Area Between Two t-scores
Video
Find t-score Given Left-Tailed Area
Video
Find t-score Given Right-Tailed Area
Video
Find t-scores Given Left-Tailed Area and Right-Tailed Area
Video
Finding z
α/2
and t
α/2
:
Finding z
α/2
Video
Note: Find z
α/2
and critical values when level of confidence is given. Normal Distribution is used.
Finding t
α/2
Video
Note: Find t
α/2
and critical values when level of confidence and degrees of freedom (DF) or sample size (n) are given.
t-Distribution is used.
Construction of Confidence Interval:
Confidence Interval for One Population Mean (μ) with Known Population Standard Deviation (σ)
Video
Note: Find confidence interval for population mean (μ) when population standard deviation (σ) is known.
Confidence Interval for One Population Mean (μ) with Unknown Population Standard Deviation (σ)
Video
Note: Find confidence interval for population mean (μ) when population standard deviation (σ) is unknown.
Confidence Interval for One Population Proportion (p)
Video
Note: Find confidence interval for population proportion (p).
Confidence Interval: Two Population Means (Assumptions: Independent Samples; σ
1
≠ σ
2
)
Video
Note: Find Confidence Interval for μ
1
- μ
2
Confidence Interval: Two Population Means (Assumptions: Independent Samples; σ
1
= σ
2
)
Video
Note: Find Confidence Interval for μ
1
- μ
2
Confidence Interval: Two Population Means (Dependent Samples)
(Matched-Pair)
Video
Note: Find Confidence Interval for the mean difference (
d̄
) of two dependent populations.
Confidence Interval: Two Population Proportions
Video
Note: Find Confidence Interval for p
1
- p
2
Margin of Error and Sample Size:
Find sample size (n) needed to estimate population mean (μ) for a specified margin of error (E)
Video
Find sample size (n) needed to estimate population proportion (p) for a specified margin of error (E)
Video
Hypothesis Testing for population mean (μ)
P-Value Approach:
Note: Program calculates p-value and test statistic.
Hypothesis Testing for One Population Mean (μ)
Video
Select an Option
If population standard deviation (σ) is KNOWN, select one of the following:
Two-Tailed Test for a Population Mean: H₀: μ = μ₀ vs. Ha: μ ≠ μ₀
Right-Tailed Test for a Population Mean: H₀: μ ≤ μ₀ vs. Ha: μ > μ₀
Right-Tailed Test for a Population Mean: H₀: μ = μ₀ vs. Ha: μ > μ₀
Left-Tailed Test for a Population Mean: H₀: μ ≥ μ₀ vs. Ha: μ < μ₀
Left-Tailed Test for a Population Mean: H₀: μ = μ₀ vs. Ha: μ < μ₀
If population standard deviation (σ) is UNKNOWN (or not given), select one of the following:
Two-Tailed Test for a Population Mean: Ho: μ = μ₀ vs. Ha: μ ≠ μ₀
Right-Tailed Test for a Population Mean: Ho: μ ≤ μ₀ vs. Ha: μ > μ₀
Right-Tailed Test for a Population Mean: Ho: μ = μ₀ vs. Ha: μ > μ₀
Left-Tailed Test for a Population Mean: Ho: μ ≥ μ₀ vs. Ha: μ < μ₀
Left-Tailed Test for a Population Mean: Ho: μ = μ₀ vs. Ha: μ < μ₀
Classical or Traditional Approach; or Critical-Value Approach:
Note: Program calculates test statistic and critical value(s).
Hypothesis Testing for One Population Mean (μ)
Video
Select an Option
If population standard deviation (σ) is KNOWN, select one of the following:
Two-Tailed Test for a Population Mean: H₀: μ = μ₀ vs. Ha: μ ≠ μ₀
Right-Tailed Test for a Population Mean: H₀: μ ≤ μ₀ vs. Ha: μ > μ₀
Right-Tailed Test for a Population Mean: H₀: μ = μ₀ vs. Ha: μ > μ₀
Left-Tailed Test for a Population Mean: H₀: μ ≥ μ₀ vs. Ha: μ < μ₀
Left-Tailed Test for a Population Mean: H₀: μ = μ₀ vs. Ha: μ < μ₀
If population standard deviation (σ) is UNKNOWN (or not given), select one of the following:
Two-Tailed Test for a Population Mean: Ho: μ = μ₀ vs. Ha: μ ≠ μ₀
Right-Tailed Test for a Population Mean: Ho: μ ≤ μ₀ vs. Ha: μ > μ₀
Right-Tailed Test for a Population Mean: Ho: μ = μ₀ vs. Ha: μ > μ₀
Left-Tailed Test for a Population Mean: Ho: μ ≥ μ₀ vs. Ha: μ < μ₀
Left-Tailed Test for a Population Mean: Ho: μ = μ₀ vs. Ha: μ < μ₀
Hypothesis Testing for population proportion (p)
P-Value Approach and Finding p-value:
Note: Program calculates p-value and test statistic.
Hypothesis Testing for One Population Proportion (p)
Video
Select an Option
Two-Tailed Test for a Population Proportion: H₀: p = p₀ vs. Ha: p ≠ p₀
Right-Tailed Test for a Population Proportion: H₀: p ≤ p₀ vs. Ha: p > p₀
Right-Tailed Test for a Population Proportion: H₀: p = p₀ vs. Ha: p > p₀
Left-Tailed Test for a Population Proportion: H₀: p ≥ p₀ vs. Ha: p < p₀
Left-Tailed Test for a Population Proportion: H₀: p = p₀ vs. Ha: p < p₀
Classical or Traditional Approach; or Critical-Value Approach:
Note: Program calculates test statistic and critical value(s).
Hypothesis Testing for One Population Proportion (p)
Video
Select an Option
Two-Tailed Test for a Population Proportion: H₀: p = p₀ vs. Ha: p ≠ p₀
Right-Tailed Test for a Population Proportion: H₀: p ≤ p₀ vs. Ha: p > p₀
Right-Tailed Test for a Population Proportion: H₀: p = p₀ vs. Ha: p > p₀
Left-Tailed Test for a Population Proportion: H₀: p ≥ p₀ vs. Ha: p < p₀
Left-Tailed Test for a Population Proportion: H₀: p = p₀ vs. Ha: p < p₀
Hypothesis Testing for Difference of Two INDEPENDENT Population Means (μ
1
- μ
2
)
Classical or Traditional Approach; or Critical-Value Approach:
Hypothesis Testing for Difference of Two INDEPENDENT Population Means (μ
1
and μ
2
)
Video
Select an Option
For Two Independent Samples with Population Standard Deviations (σ₁ and σ₂) KNOWN, select one of the following:
Two-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ ≠ μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ > μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ < μ₂
For Two Independent Samples with Population Standard Deviations (σ₁ and σ₂) UNKNOWN and Assumed Not Equal (σ₁ ≠ σ₂), select one of the following:
Two-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ ≠ μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ > μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ < μ₂
For Two Independent Samples with Population Standard Deviations (σ₁ and σ₂) UNKNOWN and Assumed Equal (σ₁ = σ₂), select one of the following:
Two-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ ≠ μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ > μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ < μ₂
P-Value Approach:
Hypothesis Testing for Difference of Two INDEPENDENT Population Means (μ
1
- μ
2
)
Video
Select an Option
For Two Independent Samples with Population Standard Deviations (σ₁ and σ₂) KNOWN, select one of the following:
Two-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ ≠ μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ ≤ μ₂ vs. Hₐ: μ₁ > μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ > μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ ≥ μ₂ vs. Hₐ: μ₁ < μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ < μ₂
For Two Independent Samples with Population Standard Deviations (σ₁ and σ₂) UNKNOWN and Assumed Not Equal (σ₁ ≠ σ₂), select one of the following:
Two-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ ≠ μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ ≤ μ₂ vs. Hₐ: μ₁ > μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ > μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ ≥ μ₂ vs. Hₐ: μ₁ < μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ < μ₂
For Two Independent Samples with Population Standard Deviations (σ₁ and σ₂) UNKNOWN and Assumed Equal (σ₁ = σ₂), select one of the following:
Two-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ ≠ μ₂
Right-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ > μ₂
Left-Tailed Test for Difference Between Two Means: H₀: μ₁ = μ₂ vs. Hₐ: μ₁ < μ₂
Hypothesis Testing for Difference of Two DEPENDENT Population Means (μ
1
- μ
2
)
Classical or Traditional Approach; or Critical-Value Approach:
Hypothesis Testing for Difference of Two DEPENDENT Population Means (μ
1
- μ
2
)
(Matched-Pair)
Video
Select an Option
Two-Tailed Test for mean difference (d̄) of two dependent populations: H₀: μ = 0 vs. Hₐ μ ≠ 0
Right-Tailed Test for mean difference (d̄) of two dependent populations: H₀: μ = 0 vs. Hₐ: μ > 0
Left-Tailed Test for mean difference (d̄) of two dependent populations: H₀: μ = 0 vs. Hₐ: μ < 0
P-Value Approach:
Hypothesis Testing for Difference of Two DEPENDENT Population Means (μ
1
and μ
2
)
(Matched-Pair)
Video
Select an Option
Two-Tailed Test for mean difference (d̄) of two dependent populations: H₀: μ = 0 vs. Hₐ μ ≠ 0
Right-Tailed Test for mean difference (d̄) of two dependent populations: H₀: μ = 0 vs. Hₐ: μ > 0
Left-Tailed Test for mean difference (d̄) of two dependent populations: H₀: μ = 0 vs. Hₐ: μ < 0
Hypothesis Testing for Difference of Two Indpendent Proportions (p
1
- p
2
)
Classical or Traditional Approach; or Critical-Value Approach
Hypothesis Testing for Difference of Two Indpendent Proportions (p
1
- p
2
)
Video
Select an Option
Two-Tailed Test for Difference Between Two Proportions: H₀: p₁ = p₂ vs. Hₐ: p₁ ≠ p₂
Right-Tailed Test for Difference Between Two Proportions: H₀: p₁ = p₂ vs. Hₐ: p₁ > p₂
Left-Tailed Test for Difference Between Two Proportions: H₀: p₁ = p₂ vs. Hₐ: p₁ < p₂
P-Value Approach
Hypothesis Testing for Difference of Two Indpendent Proportions (p
1
- p
2
)
Video
Select an Option
Two-Tailed Test for Difference Between Two Proportions: H₀: p₁ = p₂ vs. Hₐ: p₁ ≠ p₂
Right-Tailed Test for Difference Between Two Proportions: H₀: p₁ = p₂ vs. Hₐ: p₁ > p₂
Left-Tailed Test for Difference Between Two Proportions: H₀: p₁ = p₂ vs. Hₐ: p₁ < p₂
ANOVA (Analysis of Variance):
Finds F-Distribution score
Video
ANOVA (One-Way Analysis of Variance), Sheffé Test, Tukey Test
Video
ANOVA (Two-Way Analysis of Variance)
Video