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### Standard Deviation, Variance, and Coefficient of Variation of Biostatistics Data

The *standard deviation* (usually abbreviated *SD,**sd,* or just *s*) of a bunch of numbers tells you how much the individual numbers tend to differ (in either direction) from the mean. It's calculated as follows [more…]

### The Range of a Set of Numbers

The *range* of a set of values in your data is the difference between the smallest value (the *minimum* value) and the largest value (the *maximum* value):

Range [more…]

### Sample Statistics and Population Parameters

Scientists conduct experiments on limited *samples* of subjects in order to draw conclusions that (they hope) are valid for a large *population* of people. Suppose you want to conduct an experiment to determine [more…]

### Accuracy and Precision in Terms of the Sampling Distribution

The idea of a sampling distribution is at the heart of the concepts of *accuracy* and *prec**i**sion*. Imagine a scenario in which an experiment (like a clinical trial or a survey) is carried out over and over [more…]

### How to Get More Accurate Measurements in Your Data

Measurement accuracy very often becomes a matter of properly calibrating an instrument against known standards. The instrument may be as simple as a ruler or as complicated as a million-dollar analyzer [more…]

### How to Improve Sampling Precision

You improve the precision of anything you observe from your sample of subjects by having a larger sample. The *central limit theorem* (or CLT, one of the foundations of probability theory) describes how [more…]

### Comparing Averages: How Situational Differences Determine Test Methods

You may wonder why there are so many tests for such a simple task as comparing averages. Well, "comparing averages" doesn't refer to a single task; it's a broad term that can apply to a lot of situations [more…]

### How to Use Student t Tests to Compare Averages

You can run the Student t tests using typical statistical software and interpret the output produced. In this example, you'll be using the software package OpenStat. [more…]

### Centiles in Biostatistics Data

The basic idea of the median (that half of your numbers are less than the median) can be extended to other fractions besides 1/2. A centile is a value that a certain percentage of the values are less than [more…]

### How to Structure Numerical Summaries into Descriptive Tables

What do you do with the basic summary statistics that convey a general idea of how a set of numbers is distributed? Generally, when presenting your results, you pick a few of the most useful summary statistics [more…]

### Show the Distribution with Histograms

*Histograms* are bar charts that show what fraction of the subjects have values falling within specified intervals. The main purpose of a histogram is to show you how the values of a numerical value are [more…]

### Summarize Grouped Data with Bars, Boxes, and Whiskers

Sometimes you want to show how a variable varies from one group of subjects to another. For example, blood levels of some enzymes vary among the different races. Two types of graphs are commonly used for [more…]

### Confidence Interval Basics

In biostatistics, it's important to be comfortable with the basic concepts and terminology related to confidence intervals. This is an area where nuances of meaning can be tricky, and the right-sounding [more…]

### Formulas for Confidence Limits in Large Samples

Most of the approximate methods for determining confidence limits are based on the assumption that your sample statistic has a sampling distribution that's [more…]

### The Confidence Interval around a Mean

Just as the SE (standard error) formulas depend on what kind of sample statistic you're dealing with (whether you're measuring or counting something or getting it from a regression program or from some [more…]

### The Confidence Interval around a Proportion

If you were to survey 100 typical children and find that 70 of them like chocolate, you'd estimate that 70 percent of children like chocolate. What is the 95 percent confidence interval [more…]

### The Confidence Interval around a Regression Coefficient

This is one time you don't need any formulas because you shouldn't attempt to calculate standard errors or confidence intervals (CIs) for regression coefficients yourself. Any good regression program can [more…]

### The Relationship between Confidence Intervals and Significance Testing

You can use confidence intervals (CIs) as an alternative to some of the usual significance tests. To assess significance using CIs, you first define a number that measures the amount of effect you're testing [more…]

### The Concept of Error Propagation

A less extreme form of the old saying "garbage in equals garbage out" is "fuzzy in equals fuzzy out." Random fluctuations in one or more measured variables produce random fluctuations in anything you calculate [more…]

### Simple Error Propagation Formulas for Simple Expressions

Even though some general error-propagation formulas are very complicated, the rules for propagating SEs through some simple mathematical expressions are much easier to work with. Here are some of the most [more…]

### Use an Online Calculator for Complicated Error-Propagation Expressions

Statpagescalculates how precision propagates through almost any expression involving one or two variables. It even handles the case of two variables with correlated fluctuations. You simply enter the following [more…]

### How to Simulate Error Propagation

Probably the most general error-propagation technique is called *Monte-Carlo analysis*. You can use this technique to solve many difficult statistical problems. Calculating how SEs propagate through a formula [more…]

### How to Compare Sets of Matched Numbers

The unpaired (independent-sample) t tests, one-way ANOVA, ANCOVA, and their nonparametric counterparts deal with comparisons between two or more groups of [more…]

### How to Compare within-Group Changes between Groups

Comparing within-group changes between groups is a special situation, but one that comes up very frequently in analyzing data from clinical trials. Suppose you're testing several arthritis drugs against [more…]

### The Basic Idea of an Analysis of Variance (ANOVA)

The so-called “one-way analysis of variance” (ANOVA) is used when comparing three or more groups of numbers. When comparing only two groups (A and B), you test the difference [more…]