Mean, median and mode
- The mean = total of the values ÷ number of values. The mean of 3, 5, 7, 9 and 11 is 35 ÷ 5 = 7.
- The median is the middle value when the data is in order. For n values, it is at position (n + 1) ÷ 2, so for 25 values it is the 13th.
- The mode is the most common value. There can be more than one mode, or none.
Averages from tables
- From a frequency table: mean = Σfx ÷ Σf. For x = 0, 1, 2, 3 with f = 4, 6, 8, 2, Σfx = 28 and Σf = 20, so the mean is 1.4.
- For grouped data, estimate the mean using class midpoints. It is only an estimate because the exact values are unknown.
- The modal class is the class with the highest frequency. Find the class containing the median by counting to the (n + 1) ÷ 2 position.
Weighted and geometric means
- A weighted mean gives some values more importance. With coursework worth 40% (score 70) and an exam worth 60% (score 80): 0.4 × 70 + 0.6 × 80 = 28 + 48 = 76.
- The geometric mean of n values is the nth root of their product. It is used for growth rates. The geometric mean of 4 and 9 is √36 = 6.
Choosing an average
- The mean uses all the data but is affected by extreme values.
- The median isn't affected by extreme values, so it is better for skewed data such as incomes.
- The mode is the only average for qualitative data, and shows the most popular choice.
Key terms
- Mean
- The total of the values divided by the number of values.
- Median
- The middle value when data is in order.
- Mode
- The most common value.
- Modal class
- The class with the highest frequency.
- Weighted mean
- A mean where some values count more than others.
- Geometric mean
- The nth root of the product of n values.
- Σ
- The symbol meaning "the sum of".