This unit starts with data that arrive in classes. The mean becomes an estimate with limits that can be stated, and a cumulative frequency curve turns a table into something any quartile or percentile can be read from. The quartiles then give a rule for outliers.
The standard deviation is built from deviations, extended to frequency tables and grouped data, and given a second form that is proved equal to the first. The next lesson is an investigation: what adding or multiplying every value does to each measure, and why.
The last two lessons use the standard deviation as a ruler. On a bell-shaped set it fixes what share of values lies within one, two and three steps of the mean, and for a sample it needs the divisor n − 1 to estimate the population.
A graphic display calculator (GDC) is optional all the way through. Lessons 1, 3, 4, 5 and 7 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Estimating from grouped data The estimated mean from mid-interval values, the modal class and the median class, and how far the true mean can be from the estimate. Slide the values across their classes and watch the mean move between its least and greatest possible values. GDC optional
- Lesson 2 · Cumulative frequency, quartiles and percentiles Cumulative frequency plotted at upper boundaries, the median, quartiles and percentiles by reading and by interpolating, and the outlier fences. Choose a percentage of the values and read across the cumulative frequency curve to the value it gives. No calculator
- Lesson 3 · Standard deviation: the typical distance from the mean The mean as the balance point, deviations and why they are squared, the variance and the standard deviation by hand, and comparing the spread of two sets. Choose a data set and see its deviations, their squares, the variance and the standard deviation. GDC optional
- Lesson 4 · Standard deviation from tables, and the summary form The standard deviation from a frequency table and from grouped data, the form with the mean of the squares minus the square of the mean, and its proof. Choose a frequency table and a form of the formula and see that both reach the same variance. GDC optional
- Lesson 5 · Shifting and scaling data The effect of adding a constant and of multiplying by a constant on the mean, median, range, IQR and standard deviation, and the proof from the formula. Shift and scale seven marks and watch which measures follow the change and which ignore it. GDC optional
- Lesson 6 · The bell curve and standard deviations The normal distribution as a symmetric bell shape, the 68, 95 and 99.7 shares within one, two and three standard deviations, and comparing results by standard deviations from the mean. Choose a number of standard deviations and the size of one, and see the interval and the share of values it holds. No calculator
- Lesson 7 · Samples, populations and the divisor n − 1 The difference between a sample and a population, the mean and the divisor n minus 1 as estimates, the capture-recapture estimate, and what makes a sample trustworthy. Sample a small population in every possible way and compare the two divisors with the population's own variance. GDC optional
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.