Statistics Notes

Data are observations collected to answer a question. Before calculating, identify what was measured, its unit and how the observations were selected. A survey of only your friends may not represent every student in a school.

For the values 2, 2, 3, 5, 5, 5, a frequency table records value 2 with frequency 2, value 3 with frequency 1, and value 5 with frequency 3. The sum of frequencies is 6, the number of observations, not the number of distinct values.

The mean is the sum of all observations divided by their count. For 2, 2, 3, 5, 5, 5, the total is 22 and the mean is 22/6 = 11/3. The mean need not be a value in the data.

With a frequency table, multiply each value by its frequency, add these products, then divide by the total frequency. Do not average the distinct values alone; this would wrongly give rare and frequent values equal weight.

Arrange observations from smallest to largest. For an odd count n, the median is the value at position (n + 1)/2. For an even count, average the values at positions n/2 and n/2 + 1.

For 9, 2, 7, 3, 5, the sorted list is 2, 3, 5, 7, 9 and the median is 5. Adding 11 gives six values, whose middle pair is 5 and 7, so the median is 6. Positions identify which observations to use; they are not themselves the median.

The mode is a value with the highest frequency. The list 1, 2, 2, 3, 3, 8 has two modes: 2 and 3. When no value repeats, elementary summaries usually report no mode. State your convention rather than forcing a single answer.

Compare 10, 11, 12, 13, 14 with 10, 11, 12, 13, 104. The mean rises from 12 to 30 while the median stays 12. A very large observation can pull the mean away from a typical value. Do not delete an extreme value merely because it is inconvenient; investigate whether it is valid.

Use the mean when every numerical observation should contribute to the summary, the median when the centre should resist extreme values, and the mode to identify the most frequent category or value.

A bar graph compares categories using separated bars. A histogram represents grouped continuous numerical data with adjacent intervals. Read the horizontal scale carefully: a bar for marks from 10 to below 20 represents an interval, not a single score of 10.

For equal-width histogram intervals, bar heights can show frequencies. If widths differ, use frequency density (frequency divided by class width) so bar area represents frequency. Drawing raw frequencies as heights with unequal widths misrepresents the data.

Always label axes and units. Explain what the graph shows without claiming causation from a visual association alone. A cropped vertical axis can exaggerate a difference, so inspect the scale before interpreting it.

Check that the frequencies add to the number of observations. Sort the data before finding a median. For an even count, average the two middle values, not their positions. A data set can have more than one mode.

A mean of group means is valid without weighting only when group sizes are equal. If one class of 20 has mean 60 and another of 30 has mean 70, the combined mean is (20 × 60 + 30 × 70)/50 = 66, not 65.