LibGuides: Mathematics and Statistics: Explore Statistics

Learn About Key Terms

Statistics is a vocabulary-heavy discipline, full of words with highly specialized definitions (really, jargon!). Here are some key terms and definitions to get you started in statistics.

📊 Hypothesis Testing

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will:

Set up two contradictory hypotheses.
Collect sample data (in homework problems, the data or summary statistics will be given to you).
Determine the correct distribution to perform the hypothesis test.
Analyze sample data by performing the calculations that ultimately will allow you to reject or decline to reject the null hypothesis.
Make a decision and write a meaningful conclusion.

The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.

H₀: The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H_a: The alternative hypothesis: It is a claim about the population that is contradictory to H₀ and what we conclude when we reject H₀. This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H₀" if the sample information favors the alternative hypothesis or "do not reject H₀" or "decline to reject H₀" if the sample information is insufficient to reject the null hypothesis.

➗ Average or Mean?

Average or Mean are both words used to refer to a number that describes the central tendency of the data; there are several specialized averages or means, including the arithmetic mean, weighted mean, median, mode, and geometric mean.

🎲 P-value

The p-value is the probability that an event will happen purely by chance, assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.

📉 Standard deviation

The standard deviation of a data set is a number that is equal to the square root of the variance and measures how far data values are from their mean. The notation differs for sample versus population: s for sample standard deviation and σ for population standard deviation.

Statistical Significance

"In research, statistical significance measures the probability of the null hypothesis being true compared to the acceptable level of uncertainty regarding the true answer. " Read more.

References

Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

Tenny, S., & Abdelgawad, I. (2023). Statistical Significance. In StatPearls. StatPearls Publishing.

How to use this page: Each card shows your definition, then a quick “plain-language” note and an optional example. Click the example row to expand it. Use “See also” to hop to related terms.

Hypothesis Testing Core term

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will:

Set up two contradictory hypotheses.
Collect sample data (in homework problems, the data or summary statistics will be given to you).
Determine the correct distribution to perform the hypothesis test.
Analyze sample data by performing the calculations that ultimately will allow you to reject or decline to reject the null hypothesis.
Make a decision and write a meaningful conclusion.

Plain language: Treat it like a courtroom: H₀ says “nothing’s going on,” Hₐ says “something’s different.” You check your data (evidence) and decide if there’s enough to reject H₀.

See a quick example

A battery brand claims an average life of 10 hours (H₀: μ = 10). Your sample suggests much less. If the evidence is strong, you reject H₀ and conclude the claim is likely not true.

See also: P-value, Statistical Significance, Null Hypothesis (H₀), Alternative Hypothesis (Hₐ)

P-value Core term

The p-value is the probability that an event will happen purely by chance, assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.

Plain language: If H₀ were true, how surprising would our results be? A very small p-value = “whoa, that’d be rare,” which counts as evidence against H₀.

See a quick example

Flip a coin 10 times and get 9 heads. If the coin were fair, that result is unlikely (small p-value) — evidence the coin might be biased.

See also: Hypothesis Testing, Statistical Significance

Standard Deviation

The standard deviation of a data set is a number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation; σ for population standard deviation.

Plain language: A “spread” meter: low SD = scores close together; high SD = scores spread out.

See a quick example

Class A scores: 80–84 (tight cluster → low SD). Class B scores: 50, 70, 90, 100 (wide spread → high SD).

See also: Variance, Mean

Hypothesis Testing Core term

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will: (1) set up two contradictory hypotheses; (2) collect sample data; (3) determine the correct distribution to perform the hypothesis test; (4) analyze sample data by performing the calculations that ultimately will allow you to reject or decline to reject the null hypothesis; (5) make a decision and write a meaningful conclusion. The actual test begins by considering two hypotheses. They are called the null hypothesis (H₀) and the alternative hypothesis (H_a). Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject H₀ or not. After you have determined which hypothesis the sample supports, you make a decision: “reject H₀” or “do not reject H₀.”

Plain language: Treat it like a courtroom: H₀ says “nothing’s going on,” H_a says “something’s different.” Look at your data (evidence) and decide if there’s enough to reject H₀.

See a quick example

A battery brand claims an average life of 10 hours (H₀: μ = 10). Your sample suggests much less. If the evidence is strong, you reject H₀ and conclude the claim is likely not true.

See also: P-value, Statistical Significance, Null Hypothesis (H₀), Alternative Hypothesis (H_a)

P-value Core term

The p-value is the probability that an event will happen purely by chance, assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.

Plain language: If H₀ were true, how surprising would our results be? A very small p-value = “that’d be rare,” which counts as evidence against H₀.

See a quick example

Flip a coin 10 times and get 9 heads. If the coin were fair, that result is unlikely (small p-value) — evidence the coin might be biased.

See also: Hypothesis Testing, Statistical Significance

Standard Deviation

Plain language: A “spread” meter: low SD = scores close together; high SD = scores spread out.

See a quick example

Class A scores: 80–84 (tight cluster → low SD). Class B scores: 50, 70, 90, 100 (wide spread → high SD).

See also: Variance, Mean