STATBELT Utility - the Statistical Basics Exploratory Learning Tool

The data sheet and the menus

Main page

The sheet holds up to 25 columns and 2,500 rows. Each column has a name at the top. Every graph, statistic and test that uses data reads it from this sheet. The five menus (Data, Descriptive Statistics, Graphs, Calculators and Tests) open the tools. The Textbook tab links to the course textbook.

How to use it

1. Click a cell and type, or paste data copied from a spreadsheet.
2. Click a column name at the top and type to rename it.
3. To bring in whole columns with their names, copy them (names in the first row) from a spreadsheet, click a column name at the top, and paste. The names replace Column A, Column B and so on, and the data fills the cells below. The columns you paste into are replaced.
4. Use the arrow keys to move from cell to cell, as in a spreadsheet. Typing in a cell you moved to replaces its contents. To edit part of a value, double-click it or press F2, and the left and right arrows then move through the text.
5. Hold Shift and press the arrow keys to select a block of cells. Then press Ctrl+C (⌘C on a Mac) to copy, or Delete to clear.
6. Drag the right edge of a column name to change the column width. Double-click that edge to reset it.
7. To try the tools without your own data, open the Sample Data list above the sheet and pick a data set.
8. Open a menu and choose a tool. The tool opens above the sheet. Click the × in its top corner to close it.

Example

Choosing NBA 2014-2015 from Sample Data fills the sheet with 30 teams and 12 columns (Team, Conference, W, L, Win %, PPG and more). Most examples in this guide use that data set.

Tip. Columns that hold numbers are used for numerical tools such as histograms. Columns that hold words, such as Conference, are used as categories or as a “Group by” choice.

Results windows, copying and saving

Used by most tools

Most tools show their answer in a small floating window. From that window you can copy the result into a document, save it, or drag it out of the way.

How to use it

1. Click the button at the bottom of a tool, such as Open results in a new window or Open graph in a new window.
2. Use Copy image to paste the picture into a document, Save PNG to keep it as a file, or Copy table as text to paste the numbers into a spreadsheet.
3. Where an Edit button appears (graphs and some tables), click it to change the title, labels, colors or other settings and redraw.
4. Drag the window by its title bar. You can keep several windows open at once.

Example

Open the Descriptive Statistics tool, pick two columns and open the results. The window below is what appears.

Import or export data

Data ▸ Import or export data

Loads a CSV or Excel file into the sheet, or exports what is in the sheet.

How to use it

1. Drag a .csv, .xlsx or .xls file onto the drop area, or click Browse files.
2. Set Column names to tell the tool whether the first row holds names. “Detect automatically” works in most cases.
3. For an Excel file with several sheets, choose the sheet.
4. To export, choose whether to include column names, then click Export data as CSV or Copy data.

Example

A file replaces the data now in the sheet, so export first if you want to keep the old data.

Sort a column

Data ▸ Sort a column

Puts the values of a column in order, numerically for numbers and alphabetically for text.

How to use it

1. Choose the Column.
2. Leave “Sort all columns together” checked to keep every row’s values together. Uncheck it to sort only the chosen column.
3. Click Smallest first / A to Z or Largest first / Z to A. A column of numbers sorts by value. A column of names sorts alphabetically, ignoring capital letters.

Example

With the NBA data, choose PPG and click Largest first. Golden State, with 110 points per game, moves to the top row. Choosing Team and clicking A to Z puts the teams in alphabetical order.

Duplicate a column

Data ▸ Duplicate a column

Copies a column into the next empty column. Use it to keep an original while you sort or edit a copy.

How to use it

1. Choose the Column.
2. Click Copy to an empty column.

Example

Duplicating PPG puts a copy in the first empty column, with the same name.

Group data into bins

Data ▸ Group data into bins

Builds a frequency table that counts how many values fall in each class (bin) of a numerical column.

How to use it

1. Choose the Column. Optionally choose a Group by column.
2. Leave automatic bins checked, or uncheck it and type the left endpoint and the bin width.
3. Click Open frequency table in a new window.
4. To label every value with its class, click Add a column with each value’s bin. It fills the first empty column, such as “PPG (bin)”, with the class (for example 95.0 – 98.9) for each value, using the same bins as the table.

Example

For PPG with automatic bins, the table lists each class with its frequency, relative frequency and cumulative values.

Generate sample data

Data ▸ Generate sample data

Fills a column with random values from a distribution (normal, uniform, exponential, binomial, Poisson, t or chi-square). Use it for practice or simulation.

How to use it

1. Choose the Distribution and fill in its parameters.
2. Type the Number of values and the number of Decimals.
3. Type a Random seed if you want the same values every time. Leave it blank for new values.
4. Click Generate data.

Example

Normal with mean 100, standard deviation 15, 20 values and seed 7 fills the next empty column with 20 values.

Basic Statistics

Descriptive Statistics ▸ Basic Statistics

Computes the count, mean, median, quartiles, minimum, maximum, sample and population variance and standard deviation for the columns you choose. The mode, interquartile range (IQR) and range (max − min) are also in the list; check them to add them to the results. It can also give percentiles.

How to use it

1. Check the columns to describe, or use Select all.
2. Optionally choose a Group by column to get the statistics for each group.
3. Choose the quartile method and the number of decimals.
4. Click Show results in a new window.

Example

Checking W and PPG gives n = 30 for both. W has a mean of 41 and a standard deviation of 13.4549. PPG has a mean of 100.0167 and a standard deviation of 4.2758.

Frequency Tables

Descriptive Statistics ▸ Frequency Tables

Counts how often each value or category occurs. It can also build a two-way (contingency) table for two categorical variables.

How to use it

1. Choose the Table type: one variable, two variables, or a two-way table from counts already in the sheet.
2. Choose the variable or variables.
3. For one variable, choose the order of the categories and check the extra columns you want (relative frequency, percent, cumulative).
4. Click Show table in a new window.

Example

For the NBA data, the Conference table shows 15 Eastern and 15 Western teams. For the Titanic data, Class by Survival gives the two-way table below.

Correlation

Descriptive Statistics ▸ Correlation

Finds the correlation coefficient r for every pair of the columns you choose, and can add r². The table also shows the critical value of r for your sample size, so you can see whether the correlation is significant.

How to use it

1. Check two or more numerical columns.
2. Choose the Significance level α (0.10, 0.05 or 0.01) used for the critical value.
3. Check r² if you want the coefficients of determination as well.
4. Click Show results in a new window.

Example

For W, PPG and OPP PPG, the correlation of W with PPG is 0.7484 and of W with OPP PPG is −0.5104. Teams that score more win more, and teams that allow more win less. A correlation is significant when |r| is greater than the critical r in the table.

Regression

Descriptive Statistics ▸ Regression

Fits the least-squares line y = a + bx and reports the slope, intercept, r and r², along with the critical value of r for your sample size (two-tailed, at the α you choose) so you can compare |r| to it. You can also save the residuals in the sheet.

How to use it

1. Choose the Explanatory variable (x) and the Response variable (y).
2. Choose the Significance level α for the critical value of r.
3. Click Show results in a new window.
4. Optionally click Save residuals in the spreadsheet.

Example

With PPG as x and W as y, the results window gives the equation of the line along with r and r².

Histogram

Graphs ▸ Histogram

Shows the shape of a numerical variable by counting values in bins.

How to use it

1. Choose the Column, and a Group by column if you want one histogram per group.
2. Leave bins on automatic or set the start and bin width. Choose frequency or relative frequency for the bar height.
3. Check the options you want (values on bars, mean line, median line) and pick a color scheme.
4. Click Open graph in a new window.

Example

The PPG histogram shows how the 30 teams’ points per game are distributed.

Boxplot

Graphs ▸ Boxplot

Shows the five-number summary. Whiskers reach the minimum and maximum, or the most extreme non-outliers if you show outliers.

How to use it

1. Check one or more columns. Optionally choose Group by to compare groups.
2. Check Show outliers or Show mean if you want them.
3. Click Open graph in a new window. Click a box to highlight its values.

Example

Boxplots of PPG by Conference let you compare the two conferences on one axis.

Stem and leaf plot

Graphs ▸ Stem and leaf plot

Lists every value, with the leading digits as the stem and the next digit as the leaf.

How to use it

1. Choose the Column.
2. Leave the stem digit on Automatic, or choose it yourself.
3. Click Open graph in a new window.

Example

The stem and leaf plot of PPG shows all 30 values in order.

Dot plot

Graphs ▸ Dot plot

Stacks one dot per value above a number line. It is useful for small data sets.

How to use it

1. Choose the Column and, optionally, a Group by column.
2. Check the mean and median marks if you want them.
3. Click Open graph in a new window.

Example

The dot plot of W shows how the teams’ win totals are spread out.

Time series plot

Graphs ▸ Time series plot

Plots one or more columns against time, with the points joined by lines.

How to use it

1. Choose the Time column (years, dates or counts).
2. Check the columns to plot.
3. Click Open graph in a new window.

Example

With the Unemployment Rates data, Year on the horizontal axis and the Black and White columns checked shows both series on one graph. Both rise sharply in 2020.

Q-Q plot (normal)

Graphs ▸ Q-Q plot (normal)

Compares the sorted data to what a normal distribution would give. Use it to judge whether a numerical variable is approximately normal.

How to use it

1. Choose the Column and, optionally, a Group by column.
2. Leave the reference line on.
3. Click Open graph in a new window. Open the What to look for box for guidance.

Example

If the points stay close to the line, the data are approximately normal. Curves, steps or points far from the line suggest skewness or outliers.

Scatterplot

Graphs ▸ Scatterplot

Plots one numerical variable against another so you can see whether they are related.

How to use it

1. Choose the Explanatory variable (x) and the Response variable (y).
2. Optionally choose a Group by column.
3. Check the regression line and the trend options you want.
4. Click Open graph in a new window.

Example

PPG on x and W on y shows a clear upward trend, matching the correlation of 0.7484.

Bar graph

Graphs ▸ Bar graph

Shows how many observations fall in each category.

How to use it

1. Choose Raw data (one row per observation) or Summarized frequencies from the sheet.
2. Choose the column, the bar height, and the order of the bars. A descending order gives a Pareto chart.
3. To compare groups, choose a Group by column. A Bar arrangement choice then appears: split bars (side by side) or stacked bars.
4. Click Open graph in a new window.

Example

For the Titanic data, Class in descending order shows that the crew was the largest group. To compare survival within each class, set Group by to Survival and Bar arrangement to Stacked bars. Each class becomes one bar split into Alive and Dead, and checking “Show the value above each bar” puts the counts inside the segments. Choose Split bars instead to put Alive and Dead side by side.

Tip. Stacked bars show the total for each category. Split bars make it easier to compare the groups with each other. The Show list also offers percents within categories or within groups.

Pie chart

Graphs ▸ Pie chart

Shows each category as a slice of the whole.

How to use it

1. Choose Raw data or Summarized frequencies, then the column.
2. Choose what to show on the slices (frequency, relative frequency or percent) and the order.
3. Click Open chart in a new window.

Example

The Class pie chart with percents shows each class as a share of the 2,201 people.

Custom Discrete Probability Calculator

Calculators ▸ Custom Discrete Probability Calculator

Works with a probability distribution that you type into the sheet. It checks that the probabilities add to 1 (within 0.001) and gives the mean, variance, standard deviation and probabilities. Number boxes also accept simple expressions such as 42/sqrt(35).

How to use it

1. Type the values of X in one column and their probabilities in another. Probabilities can be decimals or fractions such as 1/6.
2. Choose those columns in X values in and Probabilities in.
3. Choose what to Find and type the x value (or a and b). To see which symbol a wording such as “at least” means, click Open common wordings for inequalities in a new window under Decimals.

Example

With x = 0, 1, 2, 3, 4 and probabilities 0.1, 0.2, 0.4, 0.2, 0.1, the mean is 2, the standard deviation is 1.0954, and P(X = 2) = 0.4.

Tip. The Find list offers P(X = x), ≤, <, ≥, >, between a and b, and all five at once.

Binomial Calculator

Calculators ▸ Binomial Calculator

Finds binomial probabilities for n independent trials with success probability p. Number boxes also accept simple expressions such as 42/sqrt(35).

How to use it

1. Type the Number of trials n and the Probability of success p. You can type p as a decimal or a fraction such as 1/3.
2. Choose what to Find and type x. If a problem is worded in words (“at least,” “no more than”), click Open common wordings for inequalities in a new window, at the bottom of the inputs, to see which symbol each wording means.
3. Check “Show steps” under the symbolic probability to see how the answer is computed. Uncheck “Show the probabilities above the bars” to see only the shape of the distribution. Check the full distribution table if you need it.

Example

With n = 10 and p = 0.5, P(X = 4) = 0.2051.

Normal Calculator

Calculators ▸ Normal Calculator

Finds areas under a normal curve, or finds the value that gives a chosen area.

How to use it

1. Type the mean μ and the standard deviation σ. Use 0 and 1 for the standard normal. The boxes also accept simple expressions such as 42/sqrt(35), which is handy for the central limit theorem (use sqrt( ) for a square root, ^ for powers, and pi).
2. Under Find a probability, choose an area to the left or right, between two values, or outside two values, then type the x value (or a and b). Or, under Find a value from an area, choose a value for an area to the left, to the right, in the middle, or in both tails, then type the area. Each choice is shown with its symbolic form, such as P(X < a) or P(X < _) = a. Choosing from one list clears the other. To see which symbol a wording such as “at least” means, click Open common wordings for inequalities in a new window under Decimals.

Example

With μ = 0 and σ = 1, the area to the left of 1.96 is 0.9750.

T-distribution Calculator

Calculators ▸ T-distribution Calculator

Finds areas under a t curve, or the t value for a given area. Number boxes also accept simple expressions such as 42/sqrt(35).

How to use it

1. Type the Degrees of freedom.
2. Under Find a probability, choose an area to the left or right, between two values, or outside two values, and type the t value(s). Or, under Find a value from an area, choose the type of area and type it. Each choice is shown with its symbolic form, such as P(X < a) or P(X < _) = a. Choosing from one list clears the other. To see which symbol a wording such as “at least” means, click Open common wordings for inequalities in a new window under Decimals.

Example

With 10 degrees of freedom, the area to the right of t = 2.228 is about 0.025.

Chi-square Calculator

Calculators ▸ Chi-square Calculator

Finds areas under a chi-square curve, or the χ² value for a given area. Number boxes also accept simple expressions such as 42/sqrt(35).

How to use it

1. Type the Degrees of freedom.
2. Under Find a probability, choose an area to the left or right, between two values, or outside two values, and type the χ² value(s). Or, under Find a value from an area, choose the type of area and type it. Each choice is shown with its symbolic form, such as P(X < a) or P(X < _) = a. Choosing from one list clears the other. To see which symbol a wording such as “at least” means, click Open common wordings for inequalities in a new window under Decimals.

Example

With 5 degrees of freedom, the area to the right of χ² = 11.07 is about 0.05.

Uniform Distribution Calculator

Calculators ▸ Uniform Distribution Calculator

Finds probabilities for a uniform distribution over a range of values. Number boxes also accept simple expressions such as 42/sqrt(35).

How to use it

1. Type the Minimum and Maximum of the range.
2. Choose what to Find; each choice is shown with its symbolic form, such as P(X < a) or P(X < _) = a. Type the value(s) a and b, or the area a. To see which symbol a wording such as “at least” means, click Open common wordings for inequalities in a new window under Decimals.

Example

On the range 0 to 10, the area to the left of 3 is 0.3.

Combinatorics

Calculators ▸ Combinatorics

Computes n!, combinations (nCr) and permutations (nPr), and shows the steps.

How to use it

1. Type n in the n! box, or n and r in the nCr or nPr boxes. Each formula has its own boxes.
2. Leave a formula’s boxes empty if you do not want that result in the output.
3. Click Show me how under a result to see the work, with cancelled factors struck through.
4. Check Include the steps if you want them in the output window.

Example

10C3 = 120 and 10P3 = 720.

How the tests are laid out

Tests menu

Every test works the same way, so once you have used one you can use them all.

How to use it

1. Where is the data? Choose Summarized data to type the sample statistics, or Column data to pick columns from the sheet.
2. Choose the type of test (left-tailed, right-tailed or two-tailed) and the significance level α.
3. Read the Test summary, the curve (the shaded area is the P-value), and the results table.
4. Check Show Decision to see whether to reject H₀ and what it means in words.
5. Click Open results in a new window or Open graph in a new window to copy or save them.
6. Click Open this data in the confidence interval option to get the matching interval without retyping.

Example

Confidence interval tools have a Show confidence interval plot check box, and sample size tools ask for a confidence level and a margin of error.

Inference for Proportions: one sample, hypothesis test

Tests ▸ Inference for Proportions ▸ One sample ▸ Hypothesis test

Tests whether a population proportion p equals a claimed value p₀.

How to use it

1. Type the number of successes x and the number of trials n (or choose column data).
2. Type the hypothesized proportion p₀, choose the type of test and α.
3. Check Show Decision.

Example

With x = 56, n = 100 and p₀ = 0.5, the test is two-tailed. The conditions are checked using np(1 − p) ≥ 10.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the population proportion is different from 0.5. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Estimate a proportion: one sample

Tests ▸ Inference for Proportions ▸ One sample ▸ Estimate a proportion

Gives a confidence interval for p. Choose the standard (z), plus-four or Wilson method.

How to use it

1. Type x and n, and the confidence level (95, 95% or 0.95).
2. Choose the method.
3. Check Show confidence interval plot to see the interval and its margin of error.

Example

With x = 56 and n = 100, the 95% interval is 0.4627 to 0.6573, with a margin of error of 0.0973.

Sample size for a proportion

Tests ▸ Inference for Proportions ▸ One sample ▸ Sample size for a margin of error

Finds the sample size needed for a chosen margin of error.

How to use it

1. Type the confidence level and the margin of error E.
2. If you have an estimate of p, type it. Leave it blank to use the most cautious value of 0.5.

Example

A 95% confidence level and a margin of error of 0.03 with no estimate of p gives n = 1067.07, so you need at least 1068.

Inference for Proportions: two samples, hypothesis test

Tests ▸ Inference for Proportions ▸ Two samples ▸ Hypothesis test

Tests whether two population proportions are equal, using the pooled proportion.

How to use it

1. Type x₁, n₁, x₂ and n₂.
2. Choose the type of test and α.
3. Check Show Decision.

Example

With 56 of 100 and 45 of 100, the results window gives z, the P-value and the decision.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the two population proportions are different. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Estimate a difference of proportions

Tests ▸ Inference for Proportions ▸ Two samples ▸ Estimate a difference of proportions

Gives a confidence interval for p₁ − p₂.

How to use it

1. Type x₁, n₁, x₂, n₂ and the confidence level.
2. Choose the method and check the plot if you want it.

Example

The interval is centered at the difference of the sample proportions, 0.11.

Sample size for a difference of proportions

Tests ▸ Inference for Proportions ▸ Two samples ▸ Sample size for a margin of error

Finds the sample size needed in each group for a chosen margin of error.

How to use it

1. Type the confidence level and the margin of error E.
2. Type estimates of p₁ and p₂ if you have them. Leave them blank to use 0.5.

Example

A 95% confidence level and a margin of error of 0.05, with no estimates of p₁ and p₂, gives 769 in each sample (1538 in all).

Inference for Means: one sample, hypothesis test

Tests ▸ Inference for Means ▸ One sample ▸ Hypothesis test

Tests whether a population mean μ equals a claimed value μ₀, with a t test (σ unknown) or a z test (σ known).

How to use it

1. Choose summarized data or column data.
2. Type μ₀, the type of test and α.
3. Choose t or z. For z, type σ.

Example

With a sample of 30 values, mean 50 and standard deviation 10, the test of μ₀ = 50 gives t = 0. The second picture uses column data: the ages of 46 U.S. presidents at inauguration tested against μ₀ = 55.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the population mean is different from 55. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Estimate a mean: one sample

Tests ▸ Inference for Means ▸ One sample ▸ Estimate a mean

Gives a confidence interval for μ with a t interval or a z interval.

How to use it

1. Type the sample mean, standard deviation and size (or choose a column).
2. Type the confidence level and choose t or z.

Example

With a mean of 50, a standard deviation of 10 and n = 30, the 95% interval is about 46.27 to 53.73.

Sample size for a mean

Tests ▸ Inference for Means ▸ One sample ▸ Sample size for a margin of error

Finds the sample size needed to estimate μ within a chosen margin of error.

How to use it

1. Type the confidence level and the margin of error E.
2. Type σ, or your best estimate of it.

Example

A 95% confidence level, E = 2 and σ = 10 give n = 96.04, so you need at least 97.

Inference for Means: two samples, hypothesis test

Tests ▸ Inference for Means ▸ Two samples ▸ Hypothesis test

Tests whether two population means differ by a hypothesized amount Δ₀, using the Welch or the pooled t test.

How to use it

1. Describe each sample in its own box (summarized or column data).
2. Type Δ₀ (usually 0), the type of test and α. Choose the method.
3. Check Show Decision.

Example

Sample 1 has a mean of 52, s = 10 and n = 30. Sample 2 has a mean of 48, s = 12 and n = 35. The Welch test gives t = 1.4657 and P = 0.1477.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the difference between the population means (μ₁ − μ₂) is different from 0. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Estimate a difference of means

Tests ▸ Inference for Means ▸ Two samples ▸ Estimate a difference of means

Gives a confidence interval for μ₁ − μ₂.

How to use it

1. Describe each sample, then type the confidence level.
2. Choose Welch or pooled.

Example

For the same two samples, the 95% interval for μ₁ − μ₂ is −1.4536 to 9.4536. It contains 0, which agrees with the test not rejecting H₀.

Sample size for a difference of means

Tests ▸ Inference for Means ▸ Two samples ▸ Sample size for a margin of error

Finds the sample size needed in each group.

How to use it

1. Type the confidence level and E.
2. Type σ₁ and σ₂, or estimates of them.

Example

With E = 3, σ₁ = 10 and σ₂ = 12, you need 105 in each sample (210 in all).

Inference for Means: paired samples, hypothesis test

Tests ▸ Inference for Means ▸ Paired samples ▸ Hypothesis test

Tests the mean of the differences when each value in one sample is paired with a value in the other.

How to use it

1. Type the mean and standard deviation of the differences, or choose two columns. Each row of the sheet is one pair, and the difference is x₁ − x₂.
2. Type the hypothesized mean difference (usually 0), the type of test and α.

Example

Using the Unemployment data, Black minus White unemployment over 31 years has a mean difference of 5.4258. The test gives t = 23.6718 and a P-value near 0.

Example conclusion: At the 0.05 significance level, there is sufficient evidence that the population mean difference is different from 0.

Estimate a mean difference: paired samples

Tests ▸ Inference for Means ▸ Paired samples ▸ Estimate a mean difference

Gives a confidence interval for the mean difference.

How to use it

1. Enter the differences’ mean and standard deviation, or choose two columns.
2. Type the confidence level.

Example

With a mean difference of 2.5, s_d = 4 and 20 pairs, the 95% interval is 0.6279 to 4.3721.

Sample size for a paired mean difference

Tests ▸ Inference for Means ▸ Paired samples ▸ Sample size for a margin of error

Finds the number of pairs needed.

How to use it

1. Type the confidence level and E.
2. Type the standard deviation of the differences.

Example

With E = 1 and a standard deviation of 4, you need at least 62 pairs.

Inference for Variation: one sample, hypothesis test

Tests ▸ Inference for Variation ▸ One sample ▸ Hypothesis test

Tests a claim about a population standard deviation or variance with a chi-square test.

How to use it

1. At the top, choose whether you are testing the standard deviation σ or the variance σ².
2. Type the sample standard deviation and size (or choose a column), then σ₀ (or σ₀²).
3. Choose the type of test and α.

Example

With s = 10, n = 25 and σ₀ = 12, the test statistic is χ² = (n − 1)s² ÷ σ₀² = 16.6667 and the P-value is 0.2749. Switching to variance converts your entries (10 becomes 100, 12 becomes 144).

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the population standard deviation is different from 12. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Tip. The test assumes the population is approximately normal. It is sensitive to departures from normality.

Estimate a standard deviation or variance

Tests ▸ Inference for Variation ▸ One sample ▸ Estimate a standard deviation or variance

Gives confidence intervals for σ and σ². The interval is not symmetric, so it has no margin of error.

How to use it

1. Type the sample standard deviation and size, and the confidence level.
2. Choose whether the plot shows σ or σ².

Example

With s = 10 and n = 25, the 95% interval for the variance σ² is 60.9693 to 193.5304, and for the standard deviation σ it is 7.8083 to 13.9115.

Inference for Variation: two samples, hypothesis test

Tests ▸ Inference for Variation ▸ Two samples ▸ Hypothesis test

Tests whether two populations have the same standard deviation (or variance) with an F test.

How to use it

1. Choose standard deviation or variance at the top.
2. Describe each sample, then choose the type of test and α.

Example

With s₁ = 10, n₁ = 25, s₂ = 8 and n₂ = 30, F = s₁² ÷ s₂² = 1.5625.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the two population standard deviations are different. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Estimate a ratio of standard deviations

Tests ▸ Inference for Variation ▸ Two samples ▸ Estimate a ratio of standard deviations

Gives a confidence interval for σ₁² ÷ σ₂² and for σ₁ ÷ σ₂.

How to use it

1. Describe each sample and type the confidence level.
2. Choose what to show on the plot.

Example

With s₁ = 10, n₁ = 25, s₂ = 8 and n₂ = 30, the 95% interval for σ₁ ÷ σ₂ is 0.8517 to 1.8614 (variance ratio 0.7254 to 3.4648). An interval that contains 1 is consistent with equal standard deviations.

Goodness-of-Fit: chi-square goodness-of-fit

Tests ▸ Goodness-of-Fit ▸ Chi-square goodness-of-fit

Tests whether a categorical variable follows a hypothesized distribution.

How to use it

1. Choose the data source. Either pick a column of categories and a column of observed counts from the sheet, or pick one column of raw data and let the tool count it.
2. Choose All categories equally likely (the proportion is 1 divided by the number of categories) or I will type the proportions. Typed proportions must add to 1.
3. Check Show Decision.

Example

With the colors Red, Blue, Green, Yellow and Brown and counts 20, 28, 24, 18 and 10, equal proportions give χ² = 9.2 with 4 degrees of freedom and P = 0.0563.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the population distribution is different from the hypothesized distribution. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

Tip. The tool warns you when an expected count is below 5.

Goodness-of-Fit: chi-square test of independence or homogeneity

Tests ▸ Goodness-of-Fit ▸ Chi-square test of independence or homogeneity

Tests whether two categorical variables are associated (independence), or whether the groups in the rows have the same distribution (homogeneity).

How to use it

1. Choose the type of test.
2. Either use a two-way table of counts in the sheet (a column of row names and one column of counts per group), or pick two columns of raw categories.
3. Read the observed, expected and contribution tables, χ², the P-value and Cramér’s V.

Example

The first picture uses the Titanic data (Class by Survival). The second uses a table typed in the sheet with groups in the rows and Yes, No and Maybe in the columns.

Example conclusion: At the 0.05 significance level, there is not sufficient evidence that the two variables are associated. This does not prove that H₀ is true; the sample simply does not provide strong enough evidence against it.

ANOVA: One-way ANOVA

Tests ▸ One-way ANOVA

Tests whether three or more population means are all equal.

How to use it

1. Choose Summarized data and type the name, mean, standard deviation and sample size of each group (2 to 8 groups). Or choose Column data and check one column per group.
2. Type α.
3. Read the group summary, the ANOVA table and the F curve.

Example

Three groups with means 82, 78 and 90, standard deviations 5, 6 and 5 and 12 values each give F = 15.6279 and P = 1.679e-5, so the means are not all equal. The tool warns you if the largest standard deviation is more than twice the smallest.

Example conclusion: At the 0.05 significance level, there is sufficient evidence that at least one population mean is different from the others.

Tip. A significant result says at least one mean differs. It does not say which one.

Color scheme

Settings ▸ Color scheme

Changes the color of the buttons, headings and graphs. If the purple is hard to see or you prefer another color, choose from purple (the default), blue, teal, green, yellow, orange, red, pink or gray. Only the hue changes, so the shading and contrast stay the same.

How to use it

1. Open the Settings menu and choose Color scheme.
2. Click a color. The whole tool changes at once.
3. Your choice is remembered on this device in the same browser.

Example

Choosing Blue turns the buttons, panel headings and the shaded parts of graphs blue. Graphs opened in a new window use the same color unless you pick a different one in the Graphs menu.