Averages and range

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Notes de leçon

What is an average?

  • An average is a single value that summarises the centre or typical value of a data set.
  • The most common average is the mean, but median and mode are also averages.
  • The three averages are often called the three Ms: mean, median and mode.
  • Averages are measures of central tendency — they show where data clusters.
  • The range is not an average; it measures how spread out the data is.

The mean

  • The mean is found by adding all values and dividing by how many there are.
  • For a list of n values, mean = sum of values ÷ n.
  • Example: for 2, 3, 4, 7, 9 the mean is (2+3+4+7+9) ÷ 5 = 25 ÷ 5 = 5.
  • The mean uses every value, so it can be pulled up or down by extreme values (outliers).

The median

  • The median is the middle value when the data is written in order.
  • If there is an odd number of values, the median is the single middle value.
  • If there is an even number of values, the median is the mean of the two middle values.
  • The median is useful when data is skewed or has outliers, because it is not affected by extreme values.
  • Example: personal income is often reported as the median so that a few very high incomes do not distort the typical value.

The mode

  • The mode is the value that appears most often in a data set.
  • A data set can have one mode, more than one mode, or no mode.
  • The mode is the only average that can be used for categorical (non-numeric) data.
  • The mode is useful when you want the most common or most frequent value.

The range

  • The range is the difference between the largest and smallest values: range = largest − smallest.
  • The range measures the spread of the data, not its centre.
  • A small range means the data values are close together; a large range means they are spread out.
  • The range is affected by outliers because it depends only on the extreme values.

Choosing the most appropriate average

  • Use the mean when the data is fairly symmetrical and you want to use all the values.
  • Use the median when the data is skewed or has outliers, so the typical value is not distorted.
  • Use the mode when the data is categorical or when the most common value is most useful.
  • The best average depends on the context and the purpose of the summary.

Finding a missing value when the mean is known

  • If you know the mean and the number of values, you can find the total: total = mean × number of values.
  • To find a missing value, subtract the sum of the known values from the total.
  • Example: if the mean of 4 numbers is 10, the total is 40. If three numbers are 8, 9 and 12 (sum 29), the missing number is 40 − 29 = 11.

Averages from a frequency table

  • In a frequency table, the mean is calculated as (sum of value × frequency) ÷ (sum of frequencies).
  • The median is the middle value when the data is listed in order, using the frequencies to count positions.
  • The mode is the value with the highest frequency.
  • The range is the largest value minus the smallest value in the table.

Averages from grouped data

  • For grouped data, you do not know the exact values, so you use the midpoint of each class interval.
  • The estimated mean is calculated as (sum of midpoint × frequency) ÷ (sum of frequencies).
  • The modal class is the class interval with the highest frequency.
  • The estimated mean is an approximation because it assumes all values in a class are at the midpoint.

Comparing two data sets

  • Compare two data sets by comparing an average (mean or median) and the range.
  • The average tells you which data set is typically higher or lower.
  • The range tells you which data set is more spread out or consistent.
  • A smaller range usually means the data is more consistent.

Diapos

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  1. 1.Which of these is the most common meaning of the word 'average' in mathematics?

    Easy
    • AThe arithmetic mean
    • BThe harmonic mean
    • CThe geometric mean
    • DThe mid-range

  2. 2.What is the median of the data set: 2, 3, 4, 7, 9?

    Easy
    • A4
    • B5
    • C7
    • D9

  3. 3.What is the mode of the data set: 2, 3, 4, 4, 7, 9?

    Medium
    • A2
    • B4
    • C5
    • D9

  4. 4.What is the range of the data set: 2, 3, 4, 7, 9?

    Medium
    • A2
    • B7
    • C9
    • D5

  5. 5.Which of the following are measures of central tendency? (Select all that apply)

    Medium
    • AMean
    • BMedian
    • CMode
    • DRange
    • EStandard deviation

  6. 6.The median is always one of the values in the data set.

    Easy

    True or false?

  7. 7.Match each term with its definition.

    Medium
    • Mean
    • Median
    • Mode
    • Range
    • The sum of values divided by the number of values
    • The middle value when data is ordered
    • The most frequent value
    • The difference between the highest and lowest values

  8. 8.Match each term to its meaning.

    Easy
    • Mean
    • Median
    • Mode
    • Range
    • The sum of the values divided by how many there are
    • The middle value when the data is put in order
    • The value that occurs most often
    • The difference between the largest and smallest values

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