Statistics and Probability: Key Formulas and Data Analysis Methods

A researcher randomly selects 5 fish from 20. What equation finds the chance of selecting a particular fish?

P(A) = favorable outcomes / total outcomes. Use basic probability.

A researcher asks for the probability that a fish is NOT infected when P(infected) = 0.15. What equation?

P(not A) = 1 - P(A). Complement rule.

A researcher records 5 stream samples: 4, 7, 5, 9, 5. What equation finds the mean?

x̄ = Σx / n. Add all observations, then divide by sample size.

A researcher has sample data and asks how spread out the observations are. Which equation finds sample variance?

s² = Σ(x − x̄)² / (n − 1). Use n − 1 for a sample.

After calculating variance, what equation finds standard deviation?

s = √s². Take the square root of the variance.

A fish is 28 cm long; the mean is 20 cm and SD is 4 cm. What equation finds its z-score?

z = (x − μ) / σ. It measures SDs from the mean.

Six astronauts are available and 2 must be selected for a crew. Order does NOT matter. What equation?

Combination: C(n,r) = n! / [r!(n−r)!].

100 runners compete for 1st, 2nd, and 3rd place. Order matters. What equation?

Permutation: P(n,r) = n! / (n−r)!.

A researcher selects 4 fish from 15. If ABCD is the same group as DCBA, what method?

Combination, because order does not matter.

A race has 10 runners and awards 1st, 2nd, and 3rd. What method?

Permutation, because the order of the winners matters.

How many ways can the letters in BANANA be arranged? What equation accounts for repeated letters?

Repeated arrangements: n! / (n₁!n₂!...). Divide by factorials for repeated letters.

A frog has a 60% chance of being male. Ten frogs are sampled. What equation finds exactly 6 males?

Binomial probability: P(X=x) = C(n,x)pˣ(1−p)ⁿ⁻ˣ.

A disease affects 10% of animals. Researchers sample 50 animals. What equation finds the expected number infected?

Binomial mean: μ = np.

A binomial problem gives n = 50 and p = 0.10. What equation finds the variance?

Binomial variance: σ² = np(1−p).

A binomial problem gives n = 50 and p = 0.10. What equation finds the standard deviation?

Binomial SD: σ = √[np(1−p)].

A researcher asks for the probability of exactly 4 successes in 12 trials. What distribution/equation?

Binomial: P(X=x) = C(n,x)pˣ(1−p)ⁿ⁻ˣ.

A researcher asks for the probability of fewer than 5 successes. What does this mean mathematically?

X < 5, so calculate P(0)+P(1)+P(2)+P(3)+P(4).

A researcher asks for the probability of 5 or fewer successes. What does this mean?

X ≤ 5, including 5.

A researcher asks for the probability of more than 8 successes. What does this mean?

X > 8, so 9 or more successes.

A researcher asks for the probability of at least 5 successes. What does this mean?

X ≥ 5, including 5.

A question asks for the probability of at least one infected animal. What strategy is often easiest?

Use the complement: P(at least 1) = 1 − P(0).

A population has two possible outcomes, a fixed n, independent trials, and constant p. What distribution?

Binomial distribution.

A distribution has mean 0 and standard deviation 1. What distribution is this?

Standard normal distribution.

A value is 2 standard deviations above the mean. What is its z-score?

z = +2.

A value is 1.5 standard deviations below the mean. What is its z-score?

z = −1.5.

Researchers manipulate drug concentration and measure algae growth. What is the independent variable?

Drug concentration; it is the variable manipulated by researchers.

Researchers manipulate drug concentration and measure algae growth. What is the dependent variable?

Algae growth; it is the variable being measured.

Researchers expose algae to 0, 1, 5, and 10 mg/L of a chemical. Which is the control?

0 mg/L, assuming it receives everything except the experimental chemical.

A treatment is assigned to an entire aquarium containing many organisms. What is the experimental unit?

The aquarium, because it independently receives the treatment.

Four independent tanks receive the same treatment. What is the number of replicates?

4 replicates. Each independently treated tank is one replicate.

Researchers randomly assign tanks to treatment groups. What design principle is this?

Randomization; it helps prevent systematic differences among treatment groups.

A researcher changes temperature along with chemical concentration. Why is this a problem?

Temperature is a potential confounding variable because it provides another explanation for treatment differences.

Researchers observe pollution and fish abundance without manipulating pollution. What type of study?

Observational study.

Researchers assign fish to different pollution treatments. What type of study?

Experimental study.

A researcher treats 3 tanks but measures 30 fish in each tank. Can the 90 fish automatically be treated as n = 90?

No. If treatment was assigned by tank, the tank is the experimental unit; n may be 3.

Data consist of species names such as bass, trout, and bluegill. What type of variable?

Categorical/qualitative.

Data consist of fish lengths such as 12.4, 15.7, and 18.2 cm. What type of variable?

Quantitative and continuous.

Data consist of the number of fish in each pond. What type of quantitative variable?

Discrete, because fish are counted in whole numbers.

Data consist of body mass in grams. Discrete or continuous?

Continuous; mass can be measured to increasingly precise values.

Data are categories with no natural ranking, such as species. What measurement scale?

Nominal.

Data are categories with an order, such as low, medium, and high. What measurement scale?

Ordinal.

Temperature measured in °C is an example of what measurement scale?

Interval, because differences are meaningful but zero is not absolute.

Body mass measured in grams is an example of what measurement scale?

Ratio, because there is a meaningful zero.

Each fish has its own row and repeated measurements are stored in separate columns. What data format?

Wide format.

Each observation has its own row, with Fish, Day, and Measurement as columns. What data format?

Long format.

Researchers measure algae biomass every day for 14 days. What graph is appropriate?

Line graph, because it shows change across an ordered variable such as time.

Researchers compare mean growth among control, low, medium, and high treatments. What graph?

Bar graph for comparing categorical treatment groups.

Researchers want to see the distribution of snail abundance across 100 quadrats. What graph?

Histogram, because snail abundance is quantitative and its distribution is being examined.

Researchers examine the relationship between temperature and dissolved oxygen. What graph?

Scatterplot, because both variables are quantitative.

Researchers compare the distributions of body mass among four species. What graph?

Boxplot, which compares medians, spread, quartiles, and potential outliers.

A sample has n = 5. When calculating sample variance, what goes in the denominator?

n − 1 = 4. Sample variance uses n − 1.

Why does sample variance use n − 1 instead of n?

Using n − 1 corrects for estimating the population mean from the sample.

What does variance measure?

Variability/spread of observations around the mean, expressed in squared units.

What does standard deviation measure?

The typical spread around the mean, expressed in the original units.

A problem asks how many different committees of 6 can be selected from 13 people. What method?

Combination, because a committee has no meaningful order.

A problem asks how many ways 3 winners can be ranked 1st, 2nd, and 3rd. What method?

Permutation, because assigning positions creates different outcomes.

A problem says "exactly 5" in a binomial setting. What probability notation?

P(X = 5).

A problem says "less than 5" in a binomial setting. What notation?

P(X < 5). Do not include 5.

A problem says "5 or less." What notation?

P(X ≤ 5). Include 5.

A problem says "more than 5." What notation?

P(X > 5). Do not include 5.

A problem says "at least 5." What notation?

P(X ≥ 5). Include 5.

A problem asks how many outcomes are possible when selecting r objects from n and order matters. What equation?

P(n,r) = n! / (n−r)!.

A problem asks how many outcomes are possible when selecting r objects from n and order does not matter. What equation?

C(n,r) = n! / [r!(n−r)!].

A binomial problem has n = 20 and p = 0.30. What is the expected number of successes?

μ = np = 20(0.30) = 6.

A binomial problem has n = 20 and p = 0.30. What is the variance?

σ² = np(1−p) = 20(0.30)(0.70) = 4.2.

A binomial problem has n = 20 and p = 0.30. What is the standard deviation?

σ = √[np(1−p)] = √4.2 ≈ 2.05.

A fish has x = 35 g, μ = 25 g, and σ = 5 g. What is its z-score?

z = (35−25)/5 = 2. The fish is 2 SD above the mean.

A sample contains 4, 7, 5, 9, and 5. What is its mean?

x̄ = (4+7+5+9+5)/5 = 6.

A sample has values 4, 7, 5, 9, and 5 with mean 6. What is the sample variance?

s² = [(−2)²+(1)²+(−1)²+(3)²+(−1)²]/4 = 4.

A sample has variance 4. What is its standard deviation?

s = √4 = 2.

A researcher wants to know whether one variable changes with another quantitative variable. What graph?

Scatterplot.

A researcher wants to see whether a variable changes across days. What graph?

Line graph.

A researcher wants to compare categories or treatment groups. What graph?

Bar graph.

A researcher wants to visualize the shape and spread of one quantitative variable. What graph?

Histogram.

A researcher wants to compare distributions across several groups. What graph?

Boxplot.