Statistical Analysis Operations Flashcards

Question 1

What is the difference between a population and a sample?

Accepted Answer

A population is the complete set of individuals or measurements that a study is interested in describing. A sample is a subset of that population selected for observation. Because measuring an entire population is often impractical, analysts use sample statistics to estimate population parameters. Good sampling aims for a representative sample.

Question 2

What does the mean measure, and when is the median preferred?

Accepted Answer

The mean is the arithmetic average, found by summing all values and dividing by the count. The median is the middle value when data are ordered. The median is preferred when data are skewed or contain outliers, because a few extreme values can pull the mean away from most observations.

Question 3

Why is standard deviation often reported alongside the mean?

Accepted Answer

Standard deviation measures how spread out values are around the mean. Reporting it alongside the mean shows both the typical value and how much individual observations vary from it. Two datasets can share the same mean yet differ greatly in spread, so the mean alone gives an incomplete picture of the data.

Question 4

What is the interquartile range, and why is it useful?

Accepted Answer

The interquartile range is the distance between the first quartile and the third quartile, which together cover the middle fifty percent of ordered data. It is useful because it describes spread without being affected by extreme values at either end. For this reason, it is paired with the median in summaries of skewed distributions.

Question 5

What does a z-score tell you about an individual value?

Accepted Answer

A z-score tells you how many standard deviations a value lies above or below the mean of its distribution. A positive z-score indicates a value above the mean, while a negative one indicates a value below it. Converting values to z-scores places different variables on a common scale, which makes comparisons across them easier.

Question 6

How do you compute the range of a dataset?

Accepted Answer

The range is computed by subtracting the smallest value from the largest value in the dataset. It gives a quick measure of total spread, but it depends entirely on two observations and ignores how the other values are distributed. For this reason, analysts pair it with robust measures like the interquartile range.

Question 7

What is the mode, and can a dataset have more than one?

Accepted Answer

The mode is the value that occurs most frequently in a dataset. Yes, a dataset can have more than one mode. A dataset with two modes is called bimodal, and one with several is multimodal. A dataset with no repeated values has no meaningful mode.

Question 8

Why should you check for outliers before computing summary statistics?

Accepted Answer

Outliers can distort summary statistics, especially the mean and standard deviation, which are sensitive to extreme values. Checking first helps you decide whether an unusual value reflects a real phenomenon, a recording error, or a data entry mistake. The response depends on the cause, and it should be documented.

Question 9

What is a frequency table, and what does it organize?

Accepted Answer

A frequency table is a summary that counts how often each value or category appears in a dataset. It organizes raw observations into a compact list, making patterns easier to see. Relative frequency tables express each count as a proportion or percentage of the total, which allows comparison between groups of different sizes.

Question 10

How does a histogram differ from a bar chart?

Accepted Answer

A histogram displays the distribution of a numerical variable by grouping values into adjacent intervals called bins, with bars touching to show continuity. A bar chart displays counts for separate categories, and its bars are usually separated by gaps. Histograms show the shape of a distribution, while bar charts compare categories.

Question 11

What does it mean for a distribution to be skewed?

Accepted Answer

A distribution is skewed when it is asymmetric, with one tail longer than the other. Right skew means the long tail extends toward larger values, and the mean usually sits above the median. Left skew means the long tail extends toward smaller values, and the mean usually sits below the median.

Question 12

What is a normal distribution, and what features define it?

Accepted Answer

A normal distribution is a symmetric, bell-shaped distribution in which the mean, median, and mode coincide at the center. Its shape is fully determined by two parameters, the mean and the standard deviation. Roughly 68 percent of values fall within one standard deviation of the mean, and about 95 percent fall within two.

Question 13

What is a box plot, and which values does it display?

Accepted Answer

A box plot is a graphical summary that displays the five-number summary of a dataset. The box spans the first quartile to the third quartile, with a line marking the median. Whiskers extend toward the smallest and largest values within a set limit, and points beyond the whiskers are plotted individually as potential outliers.

Question 14

What are the five numbers in a five-number summary?

Accepted Answer

The five-number summary consists of the minimum, the first quartile, the median, the third quartile, and the maximum of an ordered dataset. Together these values describe the center, spread, and range of the data. Because it relies on order statistics rather than arithmetic, it remains informative for skewed data with outliers.

Question 15

How does a scatter plot help when examining two numerical variables?

Accepted Answer

A scatter plot places each observation as a point, with one variable on the horizontal axis and the other on the vertical axis. It reveals the direction, strength, and form of any association, such as linear or curved patterns. It also helps reveal outliers and clusters that a single summary statistic might hide.

Question 16

What does a percentile indicate about a value in a dataset?

Accepted Answer

A percentile indicates the percentage of observations in the dataset that fall at or below that value. For example, a value at the 80th percentile is greater than or equal to 80 percent of the data. Percentiles are useful for comparing positions within a distribution.

Question 17

What is a sample space, and how is it used in probability?

Accepted Answer

A sample space is the set of all possible outcomes of a random experiment. Each individual outcome is a basic element of the sample space, and any collection of outcomes is called an event. Probabilities are assigned to events so that the probability of the whole sample space equals one.

Question 18

What are the basic rules for computing probabilities of events?

Accepted Answer

Every probability lies between zero and one, and the probabilities of all outcomes in the sample space sum to one. For mutually exclusive events, the probability of either occurring is the sum of their probabilities. The complement rule states that the probability of an event not occurring equals one minus the probability it occurs.

Question 19

What is the difference between independent and mutually exclusive events?

Accepted Answer

Two events are mutually exclusive when they cannot both occur in the same trial, so their intersection is empty. Two events are independent when the occurrence of one does not change the probability of the other. Mutually exclusive events with nonzero probabilities are always dependent.

Question 20

What is conditional probability, and how is it written?

Accepted Answer

Conditional probability is the probability of an event given that another event has already occurred. It is written P(A given B) and computed as the probability of both events divided by the probability of the conditioning event, provided that probability is not zero. It updates probabilities when new information becomes available.

Question 21

What does Bayes' theorem allow you to calculate from known probabilities?

Accepted Answer

Bayes' theorem lets you reverse a conditional probability, finding the probability of a prior event given that a later event was observed. It combines the prior probability of an event with the likelihood of the evidence to produce an updated, or posterior, probability. It supports diagnostic testing and updating beliefs.

Question 22

What is a random variable, and how do discrete and continuous types differ?

Accepted Answer

A random variable is a numerical quantity whose value depends on the outcome of a random experiment. A discrete random variable takes a countable set of values, such as the number of defective items in a batch. A continuous random variable can take any value within an interval, such as a measured length.

Question 23

What does the expected value of a random variable represent?

Accepted Answer

The expected value is the long-run average of a random variable over many repetitions of the experiment. For a discrete variable, it is computed by multiplying each possible value by its probability and summing the results. It is a measure of center for the distribution, though a single observed outcome may differ from it.

Question 24

Why do the binomial and normal distributions appear so often in applied statistics?

Accepted Answer

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same success probability. The normal distribution often approximates sums and averages of many independent variables, a result formalized by the central limit theorem. Their frequent appearance makes them central tools for modeling counts and measurements.

Question 25

What is a null hypothesis, and how does it differ from an alternative hypothesis?

Accepted Answer

The null hypothesis is a statement of no effect or no difference, which the test assumes true until evidence suggests otherwise. The alternative hypothesis states what the researcher suspects, such as a difference or an effect. Hypotheses are stated before the data are analyzed.

Question 26

What is a p-value, and how should it be interpreted?

Accepted Answer

A p-value is the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true. A small p-value indicates that the data are unlikely under the null hypothesis, which is evidence against it.

Question 27

What is the significance level, and how is it chosen?

Accepted Answer

The significance level, often written alpha, is the threshold probability of rejecting a true null hypothesis that the researcher is willing to accept. It is chosen before the data are analyzed. Common choices are small values such as 0.05, though the appropriate level depends on the field and the consequences of error.

Question 28

What are Type I and Type II errors in hypothesis testing?

Accepted Answer

A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected. The significance level controls the Type I error rate, while power, one minus the Type II error rate, measures the ability to detect a real effect.

Question 29

What is a confidence interval, and how should it be read?

Accepted Answer

A confidence interval is a range of plausible values for a population parameter, computed from sample data. A 95 percent interval comes from a method that, across repeated samples, captures the true parameter about 95 percent of the time. It does not give a 95 percent chance that this particular interval contains the parameter.

Question 30

What is the difference between a one-tailed and a two-tailed test?

Accepted Answer

A two-tailed test checks for a difference in either direction, so the rejection region is split between both tails of the sampling distribution. A one-tailed test checks for a difference in one specified direction, placing the entire rejection region in one tail. Choose the direction before seeing the data.

Question 31

When should you use a t-test instead of a z-test for a mean?

Accepted Answer

A t-test is generally used when the population standard deviation is unknown and must be estimated from the sample. It relies on the t distribution, which has heavier tails than the normal distribution and approaches it as the sample size grows. A z-test is appropriate when the population standard deviation is known, which is rare in practice.

Question 32

Why is statistical significance not the same as practical importance?

Accepted Answer

Statistical significance indicates that an observed result is unlikely under the null hypothesis, but it does not measure how large or meaningful the effect is. A very large sample can make a trivial difference statistically significant. For this reason, reports should include effect sizes and confidence intervals to show practical importance.

Question 33

What is simple linear regression, and what does it estimate?

Accepted Answer

Simple linear regression models the relationship between one explanatory variable and one response variable using a straight line. It estimates an intercept and a slope, where the slope gives the average change in the response for a one-unit increase in the explanatory variable. The fitted line usually minimizes the sum of squared residuals.

Question 34

What does the correlation coefficient measure, and what range does it take?

Accepted Answer

The correlation coefficient measures the strength and direction of a linear association between two numerical variables. It takes values from negative one to positive one, with values near the extremes indicating strong linear relationships. A value near zero indicates little linear association, though a strong nonlinear relationship can still exist.

Question 35

What is the coefficient of determination, and how is it interpreted?

Accepted Answer

The coefficient of determination, written R squared, is the proportion of variation in the response variable explained by the regression model. It ranges from zero to one in simple linear regression with an intercept. An R squared of 0.70 means the model accounts for seventy percent of the variability in the response.

Question 36

What main assumptions underlie ordinary least squares linear regression?

Accepted Answer

The main assumptions are that the relationship is linear, that residuals are independent, that residuals have constant variance across fitted values, and that residuals are approximately normal for valid inference. The model also assumes the explanatory variables are measured without substantial error. Checking residual plots helps reveal violations of these assumptions.

Question 37

What is multicollinearity, and why does it complicate multiple regression?

Accepted Answer

Multicollinearity occurs when two or more explanatory variables in a multiple regression are highly correlated with each other. It makes coefficient estimates unstable and inflates their standard errors, so individual effects become hard to interpret. Predictions can remain accurate even when coefficients are unreliable, which is why the issue matters most for explanation.

Question 38

What is overfitting, and how can it be detected in a model?

Accepted Answer

Overfitting occurs when a model captures random noise in the training data rather than the underlying pattern, so it performs poorly on new data. It can be detected when performance on training data is much better than performance on a separate validation or test set. Cross-validation is a common check for it.

Question 39

How are missing values typically handled before running an analysis?

Accepted Answer

First, analysts determine why values are missing, since the mechanism affects which methods are valid. Common options include removing incomplete records, imputing estimates such as the mean or a model-based value, and flagging missingness as its own category. Each choice can bias results, so the method and its effects should be reported clearly.

Question 40

Why should a model be checked against new data before it is used for decisions?

Accepted Answer

A model fitted to one dataset may describe that data well while failing to generalize to future observations or different populations. Checking performance on new data provides an honest estimate of predictive accuracy and reveals overfitting. Good practice also includes documenting assumptions, limitations, and the conditions under which the model should not be applied.

Statistical Analysis Operations Flashcards

1Describing Data Basics

2Summaries and Distributions

3Probability Foundations

4Hypothesis Testing Procedures

5Regression and Applied Modeling

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