Mr Kidman's Classroom
Revise the key ideas
Choosing tables and charts
- Frequency is how often a value or category occurs. A frequency table should identify values/categories, counts and the total sample size.
- Bar charts show categorical or discrete groups with equal-width separated bars; height represents frequency. Label axes and use a consistent scale.
Separated bars for travel categories - A pictogram uses symbols for counts. State the key and use fractional symbols consistently; if one symbol means 4, a half-symbol represents 2.
- A pie chart shows proportions of a total. Sector angle = × 360°. With 12 of 30 students, the sector is 144°.
- A vertical-line chart represents ungrouped discrete numerical values. A time-series line graph connects measurements in time order; it is not the same as a scatter graph.
Ungrouped data show each value separately rather than combining values into class intervals.
- Look for misleading displays: truncated axes (axes with part of the scale left out), unequal spacing, inconsistent symbol areas and omitted categories can exaggerate a difference.
Averages and spread
- Mean = . Median is the middle value after ordering; for an even count, average the two middle values. Mode is the most frequent value; a set may have no unique mode.
- Range = largest value − smallest value. It describes spread, not the typical value, and is sensitive to extreme observations.
- Worked example: For 2, 3, 3, 4, 8, the mean is 4, median 3, mode 3 and range 6. The high value 8 raises the mean more than the median.
- Choose an average for the context. Median often describes a skewed set of incomes (with a longer tail of values on one side) better than mean because a few very high incomes can pull the mean upwards.
Frequency tables and grouped data
- For a frequency table, mean = , where x is a value and f is its frequency. Multiply each value by its frequency before adding. Σ means “sum of”.
- Worked example: Values 1, 2, 3 have frequencies 2, 5, 3. Mean = = = 2.1.
- For continuous grouped data, use each class midpoint as its representative value. The resulting mean is an estimate because exact individual values are unknown.
The midpoint is halfway between the class boundaries. For 10 ≤ x < 20, it is (10 + 20) ÷ 2 = 15.
- Worked example: Classes 0 ≤ x < 10, 10 ≤ x < 20 and 20 ≤ x < 30 have frequencies 2, 5, 3. Midpoints 5, 15, 25 give estimated mean = 16.
- For equal-width classes such as those above, the modal class has the greatest frequency. Use cumulative totals to locate the class containing the median; grouped tables alone do not give an exact median value.
- When combining groups, add their totals and counts, not simply their means. Two people averaging 10 and eight people averaging 20 have combined mean = 18.
- Explain comparisons using both average and spread. Check sample sizes, outliers (values far from the general pattern) and whether the groups were measured consistently before drawing a conclusion.
