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A California city says average emergency response time fell by fifteen percent after a dispatch change.

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Is that evidence that the change worked, or could the result reflect which calls were counted, an unusually quiet month, or a new definition of response time?

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Statistics helps separate the measured claim from the causal story.

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This is study preparation only.

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It awards no score, course credit, or transfer credit.

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The city and numbers in this episode are invented for practice.

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Start by defining the data.

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A population is the full group we want to understand.

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A sample is the part actually observed.

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If the question concerns all emergency calls in a year but the report analyzes two hundred calls from April, those two hundred calls are the sample.

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A parameter describes a population.

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A statistic describes a sample.

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The true annual mean response time is a parameter, usually unknown.

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The April sample mean is a statistic used to estimate it.

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Similar words do not make the quantities interchangeable.

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A variable records a characteristic for each observational unit.

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Response time in minutes is quantitative.

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Call type, such as medical or fire, is categorical.

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A quantitative variable can be discrete, like the number of calls, or continuous, like elapsed time.

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Clear units and definitions are part of the variable.

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"Response time" is incomplete until we know when the clock starts and stops.

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Sampling method affects what a sample can represent.

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In a simple random sample, every possible sample of a stated size has an equal chance of selection.

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A stratified sample divides the population into meaningful groups, such as call type, then samples within each group.

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A cluster sample randomly selects whole groups, perhaps dispatch zones, then studies calls in the chosen zones.

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A systematic sample might select every twentieth call after a random start.

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Convenience samples use cases that are easy to reach.

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Voluntary response samples attract people who choose to participate.

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Both can be biased.

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If a city surveys only residents who completed an online complaint form, it may miss residents without reliable internet access and residents who gave up before completion.

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An observational study records what happened without assigning treatments.

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An experiment deliberately assigns conditions.

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A city that compares response times before and after a software launch has an observational before-and-after comparison unless assignment was controlled.

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Confounding occurs when another factor changes alongside the factor of interest.

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Staffing, weather, call severity, and reporting rules can all confound the software comparison.

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Descriptive statistics organize what was observed.

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Consider five response times: eight, nine, ten, eleven, and twenty-two minutes.

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To find the mean, add them.

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The total is sixty.

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Divide by five.

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The mean is twelve minutes.

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The median is the middle ordered value, which is ten minutes.

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The unusually long twenty-two-minute call pulls the mean above the median.

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Quartiles divide ordered data into four parts.

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The interquartile range describes the spread of the middle half and resists extreme values better than the range.

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A box plot displays the median, quartiles, and potential outliers.

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Variance and standard deviation use every observation.

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Standard deviation describes a typical distance from the mean in the variable's original units.

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A standard deviation of three minutes says more directly than a variance of nine squared minutes.

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Two districts can have the same mean response time but different standard deviations.

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The less consistent district poses a different policy problem.

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A percentile states relative position.

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If a response time is at the ninetieth percentile, about ninety percent of observed times are at or below it.

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That does not mean the response was ninety percent correct or ninety percent faster.

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For public service, the median describes the middle case, while the ninetieth percentile draws attention to slower cases near the tail.

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Distributions have shape, center, and spread.

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A symmetric distribution balances around its center.

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A right-skewed distribution has a longer tail toward larger values, common when most calls are handled quickly but a few take much longer.

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In a right-skewed distribution, the mean often exceeds the median.

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Report both when one number could conceal the shape.

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Probability describes uncertainty.

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An outcome is one possible result.

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An event is a set of outcomes.

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Probability ranges from zero to one.

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A probability of zero means impossible under the model.

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One means certain under the model.

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Neither statement proves that the model matches reality.

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Suppose twenty of one hundred sampled calls exceed fifteen minutes.

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The empirical probability of a sampled call exceeding fifteen minutes is twenty divided by one hundred, or zero point two.

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That is twenty percent.

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It estimates a broader rate only if the sample and measurement process are appropriate.

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The complement of an event means the event does not occur.

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If the estimated probability of exceeding fifteen minutes is zero point two, the estimated probability of not exceeding fifteen minutes is one minus zero point two, or zero point eight.

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A probability distribution pairs possible values with their probabilities.

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For a discrete count, the probabilities must be nonnegative and total one.

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Expected value is a long-run weighted average, not a promise for the next case.

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Now examine two quantitative variables together.

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A scatter plot places one variable on each axis.

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Look for direction, form, strength, and unusual points.

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Positive association means larger values of one variable tend to accompany larger values of the other.

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Negative association means larger values tend to accompany smaller values.

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Correlation measures the strength and direction of a linear relationship.

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It ranges from negative one to positive one.

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Values near positive one indicate strong positive linear association.

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Values near negative one indicate strong negative linear association.

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Values near zero indicate little linear association, but a strong curved relationship can still exist.

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Correlation has no units and is sensitive to outliers.

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It also does not establish causation.

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Ice cream sales and heat-related calls may rise together because temperature influences both.

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Removing one unusual district can change the correlation substantially, so inspect the scatter plot before summarizing it.

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A least-squares regression line predicts an output from an input by minimizing the sum of squared residuals.

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Suppose a simple model predicts response time in minutes as eighteen minus one point five times the number of available units.

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With six units, multiply one point five by six, giving nine.

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Subtract nine from eighteen.

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The predicted response time is nine minutes.

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The slope is negative one point five minutes per additional available unit.

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The intercept, eighteen minutes at zero available units, may not be operationally meaningful.

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An intercept can be mathematically necessary without being a sensible real-world scenario.

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A residual is actual value minus predicted value.

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If the actual response took eleven minutes and the model predicted nine, the residual is positive two minutes.

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Positive means the actual value lies above the prediction.

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A curved residual pattern signals that the linear model misses structure.

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Prediction within the observed input range is interpolation.

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Prediction far beyond that range is extrapolation and carries more risk.

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If data cover two through eight available units, predicting twenty units assumes the same linear pattern continues where it was never observed.

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Return to the civic claim.

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Last quarter, two hundred calls had a mean response time of forty minutes and a median of thirty.

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This quarter, one hundred fifty calls had a mean of thirty-four and a median of thirty-two.

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The mean fell fifteen percent because six divided by forty is zero point one five.

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Yet the median rose by two minutes, and the caseload fell.

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The accurate conclusion is limited: the measured mean was lower in this quarter under the reported definition.

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To claim that dispatch software caused improvement, investigate sampling, call mix, missing data, clock definitions, staffing, weather, ordinary variation, and the distribution of times.

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One correct percentage cannot answer all those questions.

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Three questions.

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First, what is the difference between a population parameter and a sample statistic?

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Second, why might a city report both the median and the ninetieth percentile?

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Third, what does a positive residual mean?

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First answer: a parameter describes the full population, while a statistic describes the observed sample and may estimate the parameter.

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Second answer: the median shows the middle case, while the ninetieth percentile reveals performance among relatively slow cases near the upper tail.

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Third answer: the actual observed value was greater than the regression model predicted.

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Artifact: write a one-page audit of a real chart or numerical claim from a California local-government report.

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Identify the population, sample, variables, sampling risk, distribution shape if available, center, spread, and any correlation or regression claim.

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Recalculate one reported percentage, name one plausible confounder, and state the strongest conclusion the evidence supports.

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Source note: Original study audio based on public learning materials, without quoting source prose.

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See OpenStax, Introductory Statistics 2e, Chapter 1, Sampling and Data;

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Chapter 2, Descriptive Statistics;

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Chapter 3, Probability Topics;

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and Chapter 12, Linear Regression and Correlation.

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Access for free at openstax.org.

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The current OpenStax web preface identifies Introductory Statistics 2e as licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0, or CC BY-NC-SA 4.0.

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Also consult Khan Academy public Statistics and Probability lessons on study design, descriptive statistics, probability, distributions, scatter plots, correlation, least-squares regression, and residuals.

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Preparation only.

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No credit or transfer credit is awarded by this episode.