Centre from frequency displays
Frequency weights encode repeated values. Exact percentages encode relative weights. Grouped intervals discard the exact values within each class
- Separate value and frequency Read the horizontal value and vertical frequency accurately. Zero frequency means no observation at that value
- Central positions Use cumulative frequencies. For even counts, average the two central values, not their position numbers
- Exact percentages Multiply each exact value by its percentage and divide the sum by 100. \(\displaystyle \bar{x}=\frac{\sum xp}{100}\)
- Grouped data Midpoints provide an estimate, not generally the exact mean. A median class is not an exact median
Worked example
Values 1, 3 and 7 have frequencies 2, 1 and 3. Find the mean to 2 decimal places and the exact median
- \(\displaystyle n=2+1+3=6\)Count all observations
- \(\displaystyle T=1(2)+3(1)+7(3)=26\)Recover the weighted total
- \(\displaystyle \bar{x}=\frac{26}{6}\approx4.33\)Divide by the count and round only the final result
- \(\displaystyle \text{median}=\frac{3+7}{2}=5\)The ordered observations are 1, 1, 3, 7, 7, 7; central positions 3 and 4 contain 3 and 7.
Common mistakes
Difficulty levels
Questions about this topic
Why not average the row labels?+
Rows can have different frequencies. Each observation contributes once, so multiply each value by its frequency
Does a zero frequency count as an observation?+
No. But a zero data value with positive frequency counts that many times in the denominator
Can a median fall between displayed values?+
Yes. With an even count, the two central observations can differ and their average need not be observed
Do exact percentages need a sample size for a mean?+
No. Their relative weights determine the mean: sum value times percentage and divide by 100.
What if percentages are rounded?+
Their exact underlying weights may be unknown. Treat the calculated mean as an estimate unless more information establishes exactness
What if cumulative percentage is exactly 50%?+
The two central values of an even-sized dataset lie on opposite sides of that boundary. Average those values
Can I report a midpoint mean as exact?+
Not from grouped intervals alone. Explain that class midpoints were used to estimate the mean
Is the mode the tallest frequency?+
The mode is the data value with the greatest frequency. Report its value, not the height of its bar
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions