Math: Data, Chance and Counting

This is Unit Fifteen of ACT Preparation: data, chance and counting, the last mathematics unit. Unit fourteen measured figures; this unit measures data and likelihood. One habit runs through it: before computing anything, ask what the numbers describe and how far they reach. By the end of this part you will be able to recognize a biased sample. To read histograms and box plots. To find and compare measures of center, including the score needed for an average. To find quartiles and predict what a new value does. And to predict from a line of best fit, inside the data.

Three entries on how data are collected, and on bias.

Population and sample.

ACT 1401 Population and Sample

A population is the whole group a study is about; a sample is the part actually measured. A number describing a population is a parameter, and one describing a sample is a statistic. Surveying 200 of a city's 50,000 voters estimates a parameter. A conclusion reaches only as far as the population the sample represents.

Ways to choose a sample.

ACT 1402 Sampling Methods

A simple random sample gives every group of the same size an equal chance. Stratified sampling draws randomly within subgroups, cluster sampling takes whole groups, and systematic sampling takes every kth member. Five students at random from each grade is stratified; two whole classrooms is cluster. A convenience sample is not random.

Bias.

ACT 1403 Bias

A sample is biased when the way it was chosen systematically favors some outcomes. Polling gym members about exercise overstates how much the whole town exercises. A larger biased sample is still biased; size does not fix how the sample was chosen.

Three entries on reading data displays.

Bar graphs and histograms.

ACT 1404 Bar Graphs and Histograms

A bar graph compares categories; a histogram shows how numerical data are distributed, with bars over equal intervals. A histogram of test scores in tens shows where most scores fall. Read the axes and scale before the bars: a scale starting above zero exaggerates differences.

Box plots.

ACT 1405 Box Plots

A box plot shows the five-number summary: minimum, first quartile, median, third quartile and maximum, with the box holding the middle half of the data. A box from forty to seventy with a line at fifty-five has median fifty-five. A box plot does not show the mean.

Reading a display, step by step.

ACT 1406 Reading a Display

Read the title, labels, units and scale; find exactly what the question asks; then read the values, compute, and check the answer fits the scale. In a histogram of forty commutes, ten plus eight students take twenty minutes or more, forty-five percent, and since twenty-two values lie below twenty minutes, the median falls in the ten-to-nineteen interval. A histogram can place the median in an interval but cannot give its exact value.

Nine entries on center and spread.

Mean, median and mode.

ACT 1407 Mean, Median and Mode

The mean is the sum divided by the count; the median is the middle of the ordered data, or the average of the two middle values; the mode is the most frequent value. For three, five, five, eight, nine, the mean is six and the median and mode are five. Order the data first.

The weighted average.

ACT 1408 Weighted Average

A weighted average multiplies each value by its weight, adds, and divides by the total weight. Twenty students averaging eighty and thirty averaging ninety average eighty-six, not eighty-five. Averaging two averages ignores group size; it works only when the groups are equal.

The score needed, step by step.

ACT 1409 The Score Needed for an Average

Multiply the target average by the total count, including the missing value, subtract the known sum, and check the result is possible. Scores of seventy-eight, eighty-five, ninety and eighty-two sum to 335, and an average of eighty-five over five tests needs 425, so a ninety. An average of eighty-eight would need 105, impossible on a hundred-point test. Multiply by the new count, five, not four.

Outliers and skew.

ACT 1410 Outliers and Skew

An outlier pulls the mean toward it but barely moves the median, and a skewed distribution's mean is pulled toward the long tail. Salaries of forty, forty-two, forty-five and 300 thousand have a median of 43.5 thousand and a mean of 106.75 thousand. With an outlier, the median describes the typical value better.

Range and interquartile range.

ACT 1411 Range and Interquartile Range

The range is the maximum minus the minimum; the interquartile range is the third quartile minus the first. Values more than one and a half interquartile ranges past a quartile are potential outliers: quartiles of twenty and thirty-two give an interquartile range of twelve, so values below two or above fifty. The interquartile range resists outliers; the range does not.

Quartiles, step by step.

ACT 1412 Finding the Median and Quartiles

Order the data, find the median, find each quartile as the median of its half, leaving the overall median out when the count is odd, and compute the interquartile range. Seven, nine, eleven, twelve, fourteen, fifteen, twenty has median twelve, quartiles nine and fifteen, and an interquartile range of six. Quartile conventions differ slightly between books, so use one method; ordering first is never optional.

Standard deviation.

ACT 1413 Standard Deviation as Spread

The standard deviation measures how far values typically lie from the mean; larger means more spread. Forty-eight, fifty, fifty-two and twenty, fifty, eighty share a mean of fifty, but the second set has the much larger standard deviation. Compare how far values sit from the center, not how large they are.

Percentiles.

ACT 1414 Percentiles

A value at the pth percentile is at or above p percent of the data; the median is the fiftieth percentile and the quartiles the twenty-fifth and seventy-fifth. A score at the eightieth percentile meets or beats eighty percent of the scores. A percentile is a rank, not a percent correct.

What one new value does, step by step.

ACT 1415 What a New Value Does

Compare the new value with the mean to see which way the mean moves, note that the median shifts by at most half a position, and check whether the range changes. Adding thirty to four, six, seven, nine, fourteen moves the mean from eight to about 11.7, the median from seven to eight, and the range from ten to twenty-six. A value equal to the mean leaves the mean unchanged but still changes the count, and can shift the median.

Five entries on two variables: association, the line of best fit, and correlation.

Scatter plots.

ACT 1416 Scatter Plots and Association

A scatter plot shows paired data as points. If y tends to rise as x rises, the association is positive; if it tends to fall, negative; with no pattern, none. An association describes a tendency; individual points can go the other way.

The line of best fit.

ACT 1417 Line of Best Fit

A line of best fit runs through the middle of the points and models the trend, and its slope is the predicted change in y per unit of x. An invented line, predicted score equals 2.5 times hours plus sixty, predicts 2.5 more points per extra hour. It predicts typical values, not exact ones.

Inside and outside the data.

ACT 1418 Interpolation and Extrapolation

Predicting within the range of the data is interpolation; predicting beyond it is extrapolation, which is less reliable because the trend may not continue. Data for one to six hours of study supports a prediction at four hours, not at twenty, where the line predicts 110.

Correlation.

ACT 1419 Correlation

The correlation coefficient r runs from negative one to one: its sign gives the direction of a linear association, and its size the strength. Negative 0.9 is strong and negative; 0.1 is weak. Correlation is not causation: two variables can move together because a third affects both.

Predicting from a line, step by step.

ACT 1420 Predicting From a Line of Best Fit

Read the line's equation and what x and y measure, check that x lies inside the data, then substitute and interpret with units. An invented line for monthly ice-cream sales, in thousands of dollars, against temperature is 1.2x minus thirty, fitted to months from forty to ninety degrees: at seventy-five degrees it predicts about $60,000, and each extra degree goes with about $1,200 more. At twenty degrees it predicts negative sales, impossible: an extrapolation has broken the model.

Look at what you can now do. You can tell a sample from its population and recognize bias that no sample size fixes. You can read a histogram and a box plot, starting with the scale. You can find mean, median and weighted averages, and the score needed for a target. You can find quartiles, spot potential outliers, and predict what a new value does to the center. And you can predict from a line of best fit, inside the data, without mistaking correlation for cause. Part two turns to probability and counting.