BIOL 300 Midterm Lecture

L1-L14, UBC

What are the two broad goals of statistics?

(1) Estimate the values of important parameters, (2) Test hypotheses about those parameters

Define a population in statistics.

The entire collection of individual units that a researcher is interested in

Define a parameter.

A characteristic of a population

Define a sample.

A subset of individuals from the population, used to estimate parameters

Define a variable.

A characteristic measured on individuals drawn from a population under study

Define data.

Measurements of one or more variables made on a collection of individuals

What is bias?

The tendency of a measurement process to over- or under-estimate the true population value

Volunteers are recruited for a study. Why might estimates from this sample be biased?

Volunteers are likely systematically different from the overall population

Define accuracy in statistics.

Accuracy = unbiased; estimates on average are centered on the true parameter

Define precision in statistics.

Precision = consistency; estimates give similar answers repeatedly

Can a sample be precise but inaccurate? Explain.

Yes — estimates could be tightly clustered but systematically biased away from the true parameter

List three properties of a good sample.

(1) Sufficiently large, (2) Equal probability of inclusion, (3) Independent selection

If a sample is large but all individuals come from the same neighborhood, which property is violated?

Independence (and possibly representativeness of the population)

What is a response variable vs an explanatory variable?

Response = outcome measured; Explanatory = variable used to predict or explain the response

In a study on exercise and cholesterol, which is explanatory and which is response?

Explanatory = exercise, Response = cholesterol level

What is independent sampling?

The chance of an individual being included in the sample does not depend on who else is sampled, and measurement of one sample does not influence another

Give an example of non-independence in sampling.

Catching one fish disturbs the water, affecting whether others are captured

Give another example of non-independence in sampling.

Collecting data from people in the same household; their responses may be similar due to shared environment

What is a repeated measures design?

Repeatedly measuring the same individual over time, such as tracking physiological response to a drug

What is a population parameter?

A fixed constant that describes the truth about the entire population

What is a sample estimate?

A random variable that changes depending on which sample is collected

What is absolute frequency?

The raw count of how many individuals fall into a category or bin

What is relative frequency?

The proportion or percentage of individuals in a category

Define sampling error.

Random difference between a sample estimate and the population parameter, caused by chance

Define sampling bias.

Systematic error from how a sample is collected, leading estimates away from the true population value

Contrast bias vs error.

Bias = systematic tilt, Error = random noise

How does sample size affect sampling error?

Larger samples shrink sampling error but cannot eliminate it completely

What are the four reasons estimates do not equal parameters?

Measurement bias, Sampling bias, Measurement error, Sampling error

Give an example of measurement bias.

Measuring people’s height with shoes on

Give an example of measurement error.

Small random slips when using a tape measure

Give an example of sampling bias.

Sampling too many basketball players in a student height study

Why are graphs important in statistics?

They reveal patterns in data and prevent misinterpretation from numbers alone

What are the two main types of variables?

Categorical (nominal, ordinal) and numerical (discrete, continuous)

What graph is best for categorical variables?

Bar chart (most common) or pie chart for proportions

What graph is best for numerical variables?

Histogram, CDF, or boxplot

What is a histogram?

A plot of numerical data grouped into bins with bars that touch

What are quantiles?

Cut points dividing data into equal portions

What are quartiles?

Q1 = 25th percentile, Q2 = median, Q3 = 75th percentile

What does a CDF show?

The cumulative proportion of data values ≤ a given point

What is a contingency table used for?

Showing associations between two categorical variables

What is a mosaic plot?

Graphical representation of a contingency table where rectangle areas correspond to frequencies

What graph is best for comparing numerical variables between categories?

Boxplot, violin plot, or multiple histograms

What graph is best for association between two numerical variables?

Scatter plot

What graph is best for trends over time?

Line graph

What are the two pillars of describing numerical data?

Location (central tendency) and width (spread)

What are the three common measures of location?

Mean, median, mode

What is the sample mean formula?

Ȳ = (ΣYi) / n

What is the population mean symbol?

μ

Define median.

The middle value when data are ordered; splits data into two equal halves

When is the mean preferred over the median?

For symmetric, bell-shaped data without extreme outliers

When is the median preferred over the mean?

For skewed data or when outliers are present

What is mode?

The most frequent value; unimodal, bimodal, or multimodal

How do mean, median, and mode relate in a symmetric distribution?

Mean ≈ Median ≈ Mode

How do they relate in a right-skewed distribution?

Mode < Median < Mean

How do they relate in a left-skewed distribution?

Mean < Median < Mode

What are the main measures of spread?

Range, variance, standard deviation, interquartile range, coefficient of variation

Define range.

Max – Min

Define interquartile range (IQR).

Q3 – Q1, the middle 50% of the data

When is a value often considered an outlier using IQR?

If it lies more than 1.5 × IQR below Q1 or above Q3

Define variance.

The average squared deviation from the mean

What is the difference between population and sample variance formulas?

Population divides by N, sample divides by n–1

Define standard deviation.

The square root of variance; spread in the same units as data

What does the empirical rule say for a bell-shaped distribution?

~68% within 1 SD, ~95% within 2 SDs

Define coefficient of variation (CV).

SD divided by mean, expressed as a percentage

Why is CV useful?

It allows comparison of variability across datasets with different scales

What happens to the mean if a constant is added to all values?

The mean shifts by that constant

What happens to variance if a constant is added?

Variance does not change

What happens to the mean if all values are multiplied by c?

The mean is multiplied by c

What happens to the variance if all values are multiplied by c?

Variance is multiplied by c²

What two factors determine how precise a sample mean is?

Sample size (n) and population spread (σ)

How does sample size affect precision of the mean?

Larger n → less variation in sample means

How does population spread affect precision of the mean?

Larger σ → more variation in sample means

What is the most precise sampling distribution?

Large sample size and small population spread

What is the least precise sampling distribution?

Small sample size and large population spread

What is the difference between standard deviation and standard error?

SD = spread of raw data; SE = spread of sample means

Define standard error of the mean.

The standard deviation of the sampling distribution of the sample mean

What does the standard error predict?

The sampling error of the estimate

Define population distribution, sample mean, and sampling distribution.

Population distribution = all raw data; Sample mean = average of one sample; Sampling distribution = distribution of means from repeated samples

How does the mean of sample means relate to the population mean?

The mean of sample means equals the population mean (unbiased)

What is the formula for the estimated SE of the mean?

SE = s / √n

Define a 95% confidence interval.

A range of plausible values for the true population mean

Shortcut method for 95% CI?

Sample mean ± 2 SE

How does increasing n affect confidence intervals?

Bigger n → narrower CI → more precise estimate

Why can’t you say “there is a 95% probability the mean is in this CI”?

µ is fixed; probability doesn’t apply. Instead, 95% of CIs from repeated samples will capture µ

Define pseudoreplication.

Treating non-independent measurements as if they were independent

Why is pseudoreplication a problem?

It underestimates SE, leading to misleading CIs and hypothesis tests

Example of pseudoreplication.

Measuring 10 cells from 5 mice and treating them as n=50 instead of n=5

Define a proportion.

Number with attribute / total (n), range 0–1

Define probability.

Long-run relative frequency of an event if repeated infinitely

What is the difference between probability and proportion?

Probability = true population value; Proportion = estimate from a sample

Define mutually exclusive events.

Events that cannot both occur; P(A and B) = 0

Define independent events.

Events where occurrence of one gives no information about the other

What is a probability distribution?

Describes the relative frequency of all possible outcomes; probabilities sum to 1

Formula for complement rule.

P(Not A) = 1 – P(A)

How are discrete and continuous probability distributions different?

Discrete = probability at individual outcomes; Continuous = probability only defined over ranges (single values have probability 0)

In probability, what does “or” mean?

Inclusive: A, B, or both

Formula for general addition rule.

P(A or B) = P(A) + P(B) – P(A and B)

Addition rule if A and B are mutually exclusive.

P(A or B) = P(A) + P(B)

Formula for multiplication rule if A and B are independent.

P(A and B) = P(A) × P(B)

What tool can visualize conditional probability calculations?

Probability trees