Statistics and Research Methods: Population, Variables, and Data Analysis

What is a population?

The entire group of individuals/scores the researcher wants to study.

What is a sample?

A smaller group selected from the population and actually studied.

Parameter vs. statistic?

Parameter describes a population; statistic describes a sample.

What are descriptive statistics?

Methods used to organize, simplify, and summarize data.

What are inferential statistics?

Use sample data to draw conclusions about a population.

What is sampling error?

Chance differences between a sample statistic and population parameter.

Why does sampling error happen?

A sample gives an imperfect picture of its population and samples vary by chance.

What is a construct?

A concept that cannot be directly measured, like intelligence or self-esteem.

What is an operational definition?

Defines exactly how a variable/construct will be measured.

Why are operational definitions important?

They make a construct measurable and clarify exactly what the researcher means.

What is a continuous variable?

Infinitely many possible values can exist between two observed values.

What is a discrete variable?

Separate categories; no possible values exist between adjacent categories.

Nominal scale?

Labels/categories only; no meaningful order.

Ordinal scale?

Labels/categories with a meaningful order or rank.

Interval scale?

Ordered values with equal intervals, but no absolute zero.

Ratio scale?

Ordered, equal intervals, AND an absolute/true zero.

How do I remember the four measurement scales?

NOIR: Nominal, Ordinal, Interval, Ratio.

What is correlational research?

Measures two variables without manipulation; cannot establish cause-and-effect.

What makes a study experimental?

Researcher manipulates an IV, measures a DV, compares conditions, and uses control.

What is the independent variable (IV)?

The variable manipulated by the researcher — what they CHANGE.

What is the dependent variable (DV)?

The outcome variable — what the researcher MEASURES.

What is an experimental condition?

The condition/group receiving a treatment or a particular level of the IV.

What is a control condition?

Comparison condition, often receiving no treatment.

What is an extraneous variable/confound?

A variable other than the IV that could affect the DV.

Why control extraneous variables?

ensures that any observed effect on the dependent variable is genuinely caused by the independent variable rather than an outside factor. [1]

3 ways your professor controls extraneous variables?

Random assignment, matching, and using control variables.

What is a quasi-independent variable?

A preexisting group characteristic used like an IV but not manipulated.

What does Σ mean?

Summation: add the indicated values together.

What does ΣX mean?

Add all of the X scores.

ΣX² vs. (ΣX)²?

ΣX²: square each, then add. (ΣX)²: add first, then square.

Professor's order of operations?

Parentheses → squaring → multiply/divide → summation → add/subtract.

What is a frequency distribution?

An organized table showing how many scores occur in each category.

What are the two basic frequency-table elements?

X = score/category and f = frequency.

How do you calculate proportion?

p = f/N

How do you calculate percentage?

(f/N) × 100, or p × 100.

What is a grouped frequency distribution?

A table where X categories are groups/class intervals instead of individual scores.

Why use a grouped frequency distribution?

To organize data that cover a wide range of values more clearly.

What is lost when data are grouped?

Some exact information about individual scores is lost.

What is a histogram used for?

Interval or ratio numerical data; adjacent bars touch.

What is a bar graph used for?

Nominal or ordinal data; bars are separated.

What is a frequency polygon?

A graph using connected points; grouped-data points sit above interval midpoints.

How do you find a class-interval midpoint?

(lowest score + highest score) / 2

What is a symmetrical distribution?

The two sides have the same or approximately the same shape.

What is a positively skewed distribution?

Its tail extends toward the RIGHT/high scores.

What is a negatively skewed distribution?

Its tail extends toward the LEFT/low scores.

What is central tendency?

A single score used to represent the center of a distribution.

What is the mean?

The arithmetic average: add all scores and divide by number of scores.

Population mean formula?

μ = ΣX / N

Sample mean formula?

M = ΣX / n

What is the median?

The score that divides an ordered distribution exactly in half.

How do you find the median?

Order lowest→highest; find middle. If 2 middle scores, average them.

What is the mode?

The score/category occurring most frequently.

Can a distribution have multiple modes?

Yes: unimodal, bimodal, or multimodal.

When should you use the median?

When a distribution is skewed, has extreme scores, or has undetermined values.

Why use median with an extreme score?

Extreme values pull the mean, while the median is based on the middle position.

When should you use the mode?

For nominal-scale data; also consider it for discrete variables.

Positive skew: mean, median, mode order?

Mode < Median < Mean. The mean is pulled toward the right tail.

Negative skew: mean, median, mode order?

Mean < Median < Mode. The mean is pulled toward the left tail.

What is variability?

A quantitative measure of differences between scores in a distribution.

What does standard deviation tell you?

Whether scores are typically near or far from the mean.

What is range?

Difference between the largest and smallest score (or their real limits).

What is the weakness of range?

It depends only on the two extremes, so an extreme score can greatly change it.

What is a deviation score?

A score's distance from the mean: X − μ.

What is SS?

Sum of squares: the sum of the squared deviation scores.

Computational SS formula?

SS = ΣX² − (ΣX)²/N

What is population variance?

Average squared distance from the population mean: σ² = SS/N.

What is sample variance?

Estimated variability from a sample: s² = SS/(n−1).

Why does sample variance use n−1?

Samples can underestimate population variability; n−1 helps correct that bias.

What are degrees of freedom for sample variance?

df = n−1; number of sample scores free to vary after estimating the mean.

Population standard deviation formula?

σ = √(SS/N), or √σ².

Sample standard deviation formula?

s = √[SS/(n−1)], or √s².

What is an unbiased statistic?

Its average sample-statistic value equals the population parameter.

What is a biased statistic?

It consistently underestimates or overestimates the population parameter.

What happens to SD if a constant is added to every score?

SD does NOT change.

What happens to SD if every score is multiplied by a constant?

SD changes by that multiplication factor.

What is a z-score?

A score's location measured in standard-deviation units from the mean.

Population z-score formula?

z = (X − μ) / σ

What does a positive z-score mean?

The raw score is above the mean.

What does a negative z-score mean?

The raw score is below the mean.

What does z = 0 mean?

The raw score is exactly at the mean.

What does z = +2 mean?

The score is 2 standard deviations above the mean.

How do you convert z back to raw X?

X = μ + zσ

Why are z-scores useful?

They show exact relative location and allow scores from distributions to be compared.

Does converting to z-scores change distribution shape?

No. Scores are relabeled; their relative positions and shape stay the same.

What is the mean of a z-score distribution?

0

What is the SD of a z-score distribution?

1

What is a standardized distribution?

Scores transformed to have a predetermined mean and standard deviation.

How do you create a standardized distribution?

Convert each X to z, then use that z with the new predetermined mean and SD.

How are sample z-scores different?

Same idea, but use sample mean M and sample SD s: z = (X−M)/s.

Population symbols to memorize?

N = size, μ = mean, σ² = variance, σ = standard deviation.

Sample symbols to memorize?

n = size, M = mean, s² = variance, s = standard deviation.