Quick questions on Averages and measures of spread explained: O-Level E-Maths

The mean is the total of all the values divided by how many there are:

The median is the middle value when the data is arranged in order. With an odd number of values it is the single middle one; with an even number it is the mean of the two middle values. The median is not distorted by outliers.

The mode is the value that occurs most often. A data set can have more than one mode, or none if all values are different. The mode is the only average that works for non-numerical categories.

For a frequency table, the mean is the sum of (value times frequency) divided by the total frequency. The mode is the value with the highest frequency, and the median is the value at the middle position, found by counting through the frequencies.

The range measures spread as the difference between the largest and smallest values:

For a frequency table, the median is the value at the middle position, located by counting through the cumulative frequencies. With $n$ values, the median sits at position $\tfrac{n+1}{2}$ (for odd $n$) or between positions $\tfrac{n}{2}$ and $\tfrac{n}{2}+1$ (for even $n$). Build a running total of the frequencies and find which value contains that position. For $20$ matches the median is between the $10$th and $11$th values, so you count through the frequencies until the running total reaches them.

A common twist gives the mean and all but one value, and asks for the missing one. Because the mean times the count equals the total, you can recover the total and subtract the known values. If five numbers have a mean of $8$, their total is $5 \times 8 = 40$; if four of them are $6, 7, 9, 10$ (summing to $32$), the fifth is $40 - 32 = 8$. The same idea handles a missing frequency in a table, using the formula mean $=$ (sum of value times frequency) over total frequency.

Find the mean of $4, 8, 10, 14$. [1 mark]

Find the median of $2, 5, 9, 11, 20$. [1 mark]

Find the mode and range of $3, 3, 5, 7, 7, 7, 12$. [2 marks]