<u>Sample Mean</u>

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Descriptive Statistics
Probability Distributions
Sample Size Estimation
Statistical Inference
Regression and Correlation
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Sample

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Statistics

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Descriptive Statistics
Calculations
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Uploaded Data

Select one or more items from the Statistics menu.

* Note: Quartiles are calculated by excluding the median on both sides.

** Note: Skewness and kurtosis are calculated using the Type 2 method described by Joanes and Gill (1998) . Different methods may produce slightly different values.

Sample Mean

Sample Standard Deviation

Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

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Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

Histogram

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Histogram Options

Kernel Density Estimation Curve

Distribution

Contingency Table

Binomial

Poisson

Hypergeometric

Negative Binomial

Normal

Student's t

Dimension

2 x 2 2 x 3 3 x 2 3 x 3

Probabilities

Marginal Joint Union Conditional

Number of Trials (\( n\))

Probability of Success (\( p\))

Probability

\(P(X = x \))

\(P(X \leq x)\)

\(P(X \ge x)\)

\(P(X \gt x)\)

\(P(X < x)\)

\(P(x_1 \leq X \leq x_2)\)

Number of Successes (\( x\))

Number of Successes (\( x_{1}\))

Number of Successes (\( x_{2}\))

Average (\( \lambda\))

Probability

\(P(X = x \))

\(P(X \leq x)\)

\(P(X \ge x)\)

\(P(X \gt x)\)

\(P(X < x)\)

\(P(x_1 \leq X \leq x_2)\)

Number of Successes (\( x\))

Number of Successes (\( x_{1}\))

Number of Successes (\( x_{2}\))

Population Size (\( N\))

Number of Successes in the Population (\( M\))

Sample Size (\( n\))

Probability

\(P(X = x \))

\(P(X \leq x)\)

\(P(X \ge x)\)

\(P(X \gt x)\)

\(P(X < x)\)

\(P(x_1 \leq X \leq x_2)\)

Number of Successes in the Sample (\( x\))

Number of Successes in the Sample (\( x_{1}\))

Number of Successes in the Sample (\( x_{2}\))

Required Number of Successes (\( r\))

Probability of Successes (\( p\))

Probability

\(P(X = x \))

Trials

Number of failures prior to the \(r^{th}\) success

Number of Failures prior to the \(r^{th}\) success (\( x\))

Trials until (and including) the \(r^{th}\) success (\( x\))

Population Mean (\( \mu\))

Population Standard Deviation (\( \sigma\))

Type of Calculation

Probability Critical Value

Sampling Distribution of the Sample Mean

Probability

\(P(X \leq x)\) or \(P(X < x)\)

\(P(X \ge x)\) or \(P(X \gt x)\)

\(P(x_1 \leq X \leq x_2)\)

Normally Distributed Variable (\( x\))

Normally Distributed Variable (\( x_{1}\))

Normally Distributed Variable (\( x_{2}\))

Probability

\(P(\bar{X} \leq x)\) or \(P(\bar{X} < x)\)

\(P(\bar{X} \ge x)\) or \(P(\bar{X} \gt x)\)

\(P(x_1 \leq \bar{X} \leq x_2)\)

Normally Distributed Variable (\( \bar{x}\))

Normally Distributed Variable (\( \bar{x}_{1}\))

Normally Distributed Variable (\( \bar{x}_{2}\))

Sample Size (\( n\))

Quartiles \( (Q_{1}, Q_{2}, Q_{3}) \) Percentile

Percentile Value

Degrees of freedom (\( df\))

Type of Calculation

Probability Critical Value

Probability

\(P(X \leq x)\) or \(P(X < x)\)

\(P(X \ge x)\) or \(P(X \gt x)\)

\(P(x_1 \leq X \leq x_2)\)

t Distributed Variable (\( x\))

t Distributed Variable (\( x_{1}\))

t Distributed Variable (\( x_{2}\))

Probability (\( p\))

Binomial Distribution

Calculations
Probability Distribution Table
Probability Histogram

R Code

Poisson Distribution

Calculations
Probability Distribution Table

R Code

Hypergeometric Distribution

Calculations
Probability Distribution Table
Probability Histogram

R Code

R Code

R Code

Estimate

Sample Size (\(n\)) Confidence Coefficient (\(1- \alpha\))

Methodology

Inference about one sample\(\)

Inference about two samples\(\)

Inference about more than two samples (e.g. ANOVA or Kruskal-Wallis)\(\)

Inference for Categorical Data (e.g. \( \chi^2 \) test)

Parameter of Interest

Population Mean (\( \mu \))

Population Standard Deviation (\( \sigma\))

Population Proportion (\( p\))

Data

Summarized Data Enter Raw Data Upload Data

Sample Size (\( n\))

Sample Mean (\( \bar{x}\))

Is Population Standard Deviation (\( \sigma\)) known?

Known Unknown

Population Standard Deviation (\( \sigma\)) Value

Sample Standard Deviation (\( s\)) Value

Sample

Population Standard Deviation (\( \sigma\))

Known Unknown

Population Standard Deviation (\( \sigma\)) Value

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Population Standard Deviation (\( \sigma\))

Known Unknown

Population Standard Deviation (\( \sigma\)) Value

Number of Successes (\( x\))

Number of Trials (\( n\))

Data

Summarized Data Enter Raw Data Upload Data

Sample Size (\( n\))

Sample Standard Deviation (\( s\))

Sample

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Inference Type

Confidence Interval Hypothesis Testing

Confidence Level (\( 1- \alpha\))

90% 95% 99%

Significance Level (\( \alpha\))

10% 5% 1%

Hypothesized Population Mean (\( \mu_{0}\)) Value

Hypothesized Population Proportion (\( p_{0}\)) Value

Hypothesized Population Standard Deviation (\( \sigma_{0}\)) Value

Alternate Hypothesis (\( H_{a}\))

Graph Options

Graph Options

Parameter of Interest

Two Independent Populations (\( \mu_{1} - \mu_{2} \))

Wilcoxon Rank Sum Test (or the Mann-Whitney U Test)

Dependent (Paired) Populations (\( \mu_{d} \))

Wilcoxon Signed Rank Test (Paired)

Two Population Proportions (\( p_{1} - p_{2}\))

Two Population Variances (\( \sigma_{1}^2/\sigma_{2}^2 \))

Data

Summarized Data Enter Raw Data Upload Data

Sample Size 1 (\( n_{1}\))

Sample Mean 1 (\( \bar{x}_{1}\))

Sample Size 2 (\( n_{2}\))

Sample Mean 2 (\( \bar{x}_{2}\))

Are Population Standard Deviations (\( \sigma_{1}\) and \( \sigma_{2}\)) known?

Both Known Both Unknown

Population Standard Deviation 1 (\( \sigma_{1}\)) Value

Population Standard Deviation 2 (\( \sigma_{2}\)) Value

Assume Population Variances are equal (\( \sigma_{1}^2\) = \( \sigma_{2}^2\))?

Yes (Pooled) No (Welch-Satterthwaite df)

Sample Standard Deviation 1 (\( s_{1}\)) Value

Sample Standard Deviation 2 (\( s_{2}\)) Value

Sample 1

Sample 2

Are Population Standard Deviations (\( \sigma_{1}\) and \( \sigma_{2}\)) known?

Both Known Both Unknown

Assume Population Variances are equal (\( \sigma_{1}^2\) = \( \sigma_{2}^2\))?

Yes (Pooled) No (Welch-Satterthwaite df)

Population Standard Deviation 1 (\( \sigma_{1}\)) Value

Population Standard Deviation 2 (\( \sigma_{2}\)) Value

Note: CougarStats does not store, log, or share any data you upload. All uploaded files exist only for the duration of your session and are permanently deleted when the session ends.

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Column for Sample 1

Column for Sample 2

Are Population Standard Deviations (\( \sigma_{1}\) and \( \sigma_{2}\)) known?

Both Known Both Unknown

Assume Population Variances are equal (\( \sigma_{1}^2\) = \( \sigma_{2}^2\))?

Yes (Pooled) No (Welch-Satterthwaite df)

Population Standard Deviation 1 (\( \sigma_{1}\)) Value

Population Standard Deviation 2 (\( \sigma_{2}\)) Value

Data

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Sample 1

Sample 2

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Column for Sample 1

Column for Sample 2

Data

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Sample 1 (e.g. Before, Pre-Treatment, Baseline)

Sample 2 (e.g. After, Post-Treatment, Follow-Up)

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Column for Sample 1 (e.g. Before, Pre-Treatment, Baseline)

Column for Sample 2 (e.g. After, Post-Treatment, Follow-Up)

Data

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Sample 1 (e.g. Before, Pre-Treatment, Baseline)

Sample 2 (e.g. After, Post-Treatment, Follow-Up)

Note: CougarStats does not store, log, or share any data you upload. All uploaded files exist only for the duration of your session and are permanently deleted when the session ends.

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Column for Sample 1 (e.g. Before, Pre-Treatment, Baseline)

Column for Sample 2 (e.g. After, Post-Treatment, Follow-Up)

Number of Successes 1 (\( x_{1}\))

Number of Trials 1 (\( n_{1}\))

Number of Successes 2 (\( x_{2}\))

Number of Trials 2 (\( n_{2}\))

Data

\( n_1,\ n_2,\ s_1,\ s_2 \) \( n_1,\ n_2,\ s_1^2,\ s_2^2 \) Enter Raw Data

Sample Size 1 \( (n_1) \)

Sample Standard Deviation 1 \( (s_1) \)

Sample Size 2 \( (n_2) \)

Sample Standard Deviation 2 \( (s_2) \)

Sample Size 1 \( (n_1) \)

Sample Variance 1 \( (s_1^2) \)

Sample Size 2 \( (n_2) \)

Sample Variance 2 \( (s_2^2) \)

Sample 1 (e.g Class A test scores)

Sample 2 (e.g Class B test scores)

Inference Type

Confidence Interval Hypothesis Testing

Confidence Level (\( 1- \alpha\))

90% 95% 99%

Significance Level (\( \alpha\))

10% 5% 1%

Hypothesized Population Mean Difference \( (\mu_{1} - \mu_{2})_{0} \) Value

Hypothesized Population Mean Difference \( (\mu_{d})_{0} \) Value

Hypothesized Population Proportion Difference \( (p_{1} - p_{2})_{0} \) Value

Alternate Hypothesis (\( H_{a}\))

Method

Exact Normal approximation (for large samples)

Continuity correction

True False

Graph Options

Graph Options

Q-Q Plot of the Difference (d)

Graph Options

Method

Exact Normal approximation (for large samples)

Graph Options

Q-Q Plot of the Difference

Note: CougarStats does not store, log, or share any data you upload. All uploaded files exist only for the duration of your session and are permanently deleted when the session ends.

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Choose a Sheet

Hypothesis Test

One-way Analysis of Variance (ANOVA)

Kruskal-Wallis

Data Format

Values in multiple columns

Responses and factors stacked in two columns

Choose columns to conduct analysis

Response Variable

Factors

Significance Level (\( \alpha\))

10% 5% 1%

Options

Include post hoc tests

Graph Options

Data Format

Values in multiple columns

Responses and factors stacked in two columns

Choose columns to conduct analysis

Response Variable

Factors

Significance Level (\( \alpha\))

10% 5% 1%

Graph Options

Dimension

2 x 2 2 x 3 3 x 2 3 x 3

Name for Row Variable (optional):

Name for Column Variable (optional):

Hypothesis Test

Chi-Square test for independence Fisher's Exact test

with Yates continuity correction

Significance Level (\( \alpha\))

10% 5% 1%

Analysis
Graphs
Uploaded Data

Confidence Interval

Hypothesis Test

Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

* Note: Quartiles are calculated by excluding the median on both sides.

Histogram

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Histogram Options

Kernel Density Estimation Curve

Analysis
Graphs

Confidence Interval

Hypothesis Test

Analysis
Graphs
Uploaded Data

Confidence Interval

Hypothesis Test

Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

* Note: Quartiles are calculated by excluding the median on both sides.

Histogram

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Histogram Options

Kernel Density Estimation Curve

Analysis
Graphs
Uploaded Data

Confidence Interval

Hypothesis Test

Side-by-side Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

* Note: Quartiles are calculated by excluding the median on both sides.

Q-Q Plots for Sample 1 and Sample 2

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Analysis
Data with Ranks
Graphs
Uploaded Data

Hypothesis Test

Q-Q Plot of the Difference

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Analysis
Data with Calculations
Graphs
Uploaded Data

Confidence Interval

Hypothesis Test

Q-Q Plot of the Difference (d)

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Analysis
Graphs

Confidence Interval

Hypothesis Test

Analysis
Data with Ranks
Graphs
Uploaded Data

Hypothesis Test

Ranked Results by Group

Side-by-side Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

* Note: Quartiles are calculated by excluding the median on both sides.

Q-Q Plots for Sample 1 and Sample 2

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Analysis

Confidence Interval

Hypothesis Test

Analysis
Graphs
Uploaded Data

One-way Analysis of Variance (ANOVA)

Side-by-side Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

* Note: Quartiles are calculated by excluding the median on both sides.

Histogram of Residuals

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Histogram Options

Kernel Density Estimation Curve

QQ Plot of Residuals

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Group Means

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Analysis
Data table with Ranks
Graphs
Uploaded Data

Hypothesis Test

Side-by-side Boxplot

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Orientation

Plot Horizontally

Boxplot Options

Box Width

* Note: Quartiles are calculated by excluding the median on both sides.

Group Means

Plot Options

Main title and axes labels:

Plot Colour

Plot Height

auto in px

Plot Width

auto in px

Add Gridlines

Major Minor

Chi-Square Test for Independence

Observed Frequencies

Expected Frequencies

Calculation of the \( \chi^2 \) statistic value

Fisher's Exact Test

Observed Frequencies

Hypothesis Test

Methodology

Simple Linear Regression and Correlation Analysis

Multiple Linear Regression

Polynomial Regression

Binary Logistic Regression

Note: CougarStats does not store, log, or share any data you upload. All uploaded files exist only for the duration of your session and are permanently deleted when the session ends.

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Methodology

Principal Component Analysis

k-Nearest Neighbors

Linear Discriminant Analysis

Decision Trees (CART)

Random Forest

Gradient Boosting (XGBoost)

Select a categorical variable, must have 2 or more unique categories.

Response Variable

Explanatory Variables (x₁, x₂, ..., x_k)

Data Transformation

Original scale

Logarithmic transformation

Box-Cox transformation

Standardized Box-Cox transformation

Number of Factors

X-axis component

Y-axis component

Train/Test split (%)

Number of Neighbors (k)

Standardize predictors

Select a categorical variable, must have 2 or more unique categories.

Response Variable (Class)

Explanatory Variables (x₁, x₂, ..., xₖ)

Graph Options

Decision Boundary Plot

Select a categorical variable, must have 2 or more unique categories.

Response Variable (Class)

Explanatory Variables (x₁, x₂, ..., x_k)

Use Leave-One-Out Cross-Validation

Maximum Tree Depth

Minimum Split Size

Complexity Parameter

Select a categorical variable, must have 2 or more unique categories.

Response Variable (Class)

Explanatory Variables (x₁, x₂, ..., x_k)

Train/Test split (%)

Number of Trees

Variables per Split (leave blank for auto)

Select a categorical variable, must have 2 or more unique categories.

Response Variable (Class)

Explanatory Variables (x₁, x₂, ..., x_k)

Train/Test Split (%)

Number of Boosting Rounds (M)

Max Tree Depth (d)

Learning Rate (η)

Subsample Ratio (s)

Column Sample per Tree (c)

Select a categorical variable, must have 2 or more unique categories.

Response Variable (Class)

Explanatory Variables (x₁, x₂, ..., x_k)

Results
Plots
Data Transformations
Uploaded Data

Principal Component Analysis Summary

Component Loadings (Unrotated)

Component Loadings (Varimax Rotation)

Correlation Matrix

Covariance Matrix

Eigenvectors

Scree Plot

Score Plot

Biplot

Loadings Heatmap

Results
Plots
Uploaded Data

Results
Plots
Uploaded Data

Results
Plots
Uploaded Data

Results
Plots
Uploaded Data

Results
Plots
Uploaded Data