Statistics Tutoring

Identifying bias in a sampling method

A voluntary-response sample contains people who choose to participate, so strong opinions may be overrepresented. Tutors can examine the sampling frame, how participants are selected, nonresponse, and question wording separately. Equal selection chances alone do not establish that the responses represent the target population.

Finding the median with an even number of data points

With twenty ordered values, the median is the average of the tenth and eleventh values. Tutors can distinguish the count of observations, their positions in order, and the numerical values at those positions before calculating the average.

Calculating and interpreting quartiles

One common classroom method finds Q1 as the median of the lower half and Q3 as the median of the upper half. Conventions for odd-sized samples and software can differ, so tutors can follow the method specified by the task. The quartiles describe positions in ordered data, rather than individual data values' percentages.

Distinguishing conditional probability from joint probability

Conditional probability restricts the sample space to one row or column of a two-way table. The probability of working weekends given full-time employment uses only the full-time row, not the entire table. Students may divide by the wrong total if they do not identify what condition is given.

Confusing independence with mutually exclusive events

Two events are independent if knowing one does not change the probability of the other. They are mutually exclusive if both cannot happen together. Rolling a five and obtaining a sum of seven on two dice are not mutually exclusive because both can occur. Tutors can use examples to separate the two concepts.

Connect a measure of center to its meaning in context

The mean adds all values and divides by the count; the median is the middle value when data are ordered. Tutors can show both on a dot plot and discuss which measure better represents the data when an outlier is present. Students can then explain their choice rather than apply a formula without interpretation.

1. Order the data and identify the middle value or values.
2. Calculate the mean by summing all values and dividing by the count.
3. Compare the two measures and check for outliers.
4. Explain which measure answers the question in context.

Use a two-way table to clarify conditional probability

A two-way table organizes counts by two categories. To find the probability of event Y given event X, restrict attention to the row or column for X and divide the count of both events by the total for X. Tutors can highlight the restricted row and ask students to explain why the denominator changes.

Check for bias by examining the sampling method

Stratified random sampling selects within defined subgroups. A systematic sample uses a fixed interval after a starting point, often chosen randomly. Tutors can examine who appears in the sampling frame, how the interval relates to population patterns, who responds, and how questions are phrased, rather than use equal selection chances as the only test for bias.

From a data set to quartiles and outlier boundaries

Using the median-of-halves convention, order 2, 4, 6, 8, 10, 12, 14, 30. Q1 is 5 and Q3 is 13, so the IQR is 8. The 1.5-IQR fences are −7 and 25; 30 lies above the upper fence and is flagged as a potential outlier. Tutors can explain the calculation and discuss the value in context before deciding what to do with it.

From a two-way table to testing independence

Use an original finite probability model: a bag contains twenty cards, twelve blue and five marked with a star; three cards are both blue and starred. P(blue) = 12/20, P(star) = 5/20, and P(both) = 3/20. Since (12/20)(5/20) = 3/20, the two events are independent for a uniform random draw from this bag. Students can connect the calculation to the overlap in a two-way table.