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Statistics, grade 11 (4 units)

Where one value stands among all the others, and how closely two variables go together.Subjects

The course builds on the statistics and probability of grade 10 (4 units): the mean, the standard deviation, the probability of events and the probability tree. The topics, in order:

  • a recap of the mean and the standard deviation;
  • the standard score, its properties and the shapes of distributions;
  • the normal distribution: the curve’s properties, the standard normal distribution and its full table, a percent from a value and a value from a percent, finding the mean and the standard deviation, and the normal distribution with probability;
  • the relation between two quantitative variables: the scatter diagram, the means graph, the correlation coefficient, the regression line and prediction.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The mean and the standard deviation: a recapLearningThe mean and the standard deviation from a list and from a frequency table; what the standard deviation measures; what happens to both when a number is added to every value or every value is multiplied.
  2. The standard score: a value's relative positionLearningThe standard score z – how many standard deviations a value is from the mean; comparing values from different groups; from a standard score to a value; the mean and the standard deviation from two values and their standard scores.
  3. The properties of standard scoresLearningThe mean of the standard scores is 0 and their standard deviation 1; a standard score is a number without units; what happens to a standard score when a number is added to every value or every value is multiplied.
  4. The shapes of distributionsLearningA symmetric distribution, a distribution with a right tail and with a left tail, a uniform and a two-peaked distribution; where the mean and the median are in each; things that are normally distributed and things that are not.
  5. Practice: the standard score and the shapes of distributionsPracticeQuestions on the mean and the standard deviation from a frequency table, comparing by standard scores, from a standard score to a value, finding the mean and the standard deviation, the properties of standard scores and the shape of a distribution.
  6. The normal curve and its propertiesLearningThe bell curve: symmetric round the mean, the mean, median and mode equal, the area under the curve as a part of the data; the mean sets the place and the standard deviation the width; the 34/13.5/2.5 rule.
  7. The standard normal distribution and its tableLearningThe distribution of standard scores, with mean 0 and standard deviation 1; the full table of the cumulative standard normal distribution, Φ(z); the area above a standard score, below it and between two standard scores.
  8. From a value to a percentage: a part of a normal distributionLearningFrom a value to a standard score, from the table to an area: the percentage above a value, below it and between two values; the number in a range; the number tested in all from the number in a range.
  9. From a percentage to a value: reading the table backwardsLearningFrom the area to the standard score and from the standard score to the value: the boundary of the top or bottom percentages, the pass mark when the number of places is known, the middle range; comparing two exams.
  10. Finding the mean and the standard deviation from percentagesLearningWhen the percentages are known and the parameters are not: the mean when the standard deviation is known, the standard deviation when the mean is known, and both from two percentages – by a system of two equations.
  11. The normal distribution and probabilityLearningThe area under the curve as a probability; two people chosen at random – exactly one, both; two independent tests; two groups with different distributions, in a probability tree.
  12. Practice: the normal distributionPracticeQuestions on the standard normal table, a percentage from a value and a value from a percentage, finding the mean and the standard deviation, comparing tests and the normal distribution with probability.
  13. The scatter diagram and a statistical relationLearningTwo quantitative variables in one population: the scatter diagram; a positive relation, a negative relation, no relation, a perfect relation – linear or not; a statistical relation is not necessarily a causal one.
  14. The means graph and predictionLearningDividing the independent variable's axis into bands, the mean of the dependent variable in each band, and the means graph; a band's mean as the prediction; a means graph for each value of x.
  15. The correlation coefficient r: meaning and propertiesLearningThe correlation coefficient as a measure of a linear relation: the sign and the strength – strong, moderate, weak; r = ±1 and r = 0; properties: between −1 and 1, no units, unchanged by adding a number or changing units, symmetric.
  16. Working out the correlation coefficientLearningThe correlation coefficient from raw data – the mean of the products of the deviations from the means, divided by the two standard deviations – and from standardised data – the mean of the products of the standard scores; a working table.
  17. The regression line and predictionLearningThe line predicting y from x: through the point of the means, with slope r·Sy/Sx; its equation from the means, the standard deviations and the correlation coefficient; prediction with the line.
  18. Changes in the data and the limits of predictionLearningWhat happens to the regression line and the correlation coefficient when a number is added to every value of y or every value is multiplied; when prediction with the regression line misleads: a relation that is not linear, prediction far from the data, an outlier.
  19. Practice: the correlation coefficient and the regression linePracticeQuestions on the scatter diagram, choosing and computing the correlation coefficient, the regression line and prediction, and changes in the data.
  20. Review: statisticsPracticeQuestions from the whole course: the standard score, a percent from a value and a value from a percent, the mean and the standard deviation from percents, the normal distribution and probability, the correlation coefficient, the regression line and changes in the data.
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