Statistical Averages and Spreads

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Statistical Averages

3 types

Mean = Total of all values / Number of values

Median = The middle value of an ordered list

Mode = Value that occurs most frequently 

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Statistical Spread

Range = Highest value - Lowest value

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Comparing The Different Averages

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The Mean

Gives an idea of what would happen if there were equal shares.

Useful when you need to quote a typical value provided that the data is quite closely grouped around the mean.

Advantages

  • Can be calculated exactly
  • Uses all data

Disadvantages

  • Influenced by extreme/abnormal values
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The Median

Advantages

  • Simple to understand
  • Not affected by abnormal values
  • Useful when comparing with a middle value (e.g. half the class got over 50%)

Disadvantage 

  • Doesn't make use of all the data.
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The Mode

Advantages 

  • Simple to understand.
  • Not affected by abnormal values.
  • Useful to manufacturers and shops of shoes and clothes. (e.g. which shoe size is most popular)
  • Useful in opinion polls and appropriate for non-numeric data.

Disadvantages 

  • Does not make use of all the data.

 

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The Range

Useful when we want to investigate the difference between the spread of data.

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Finding Averages and the Spread from a Frequency Table

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Mean (Freq. Table)

  • Multiply each value (x) by its frequency (f) to get fx.
  • Add up the fx values to get Σfx   (Σ = Total)
  • Add up all the f values to get Σf 

Mean = Σfx Σf

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Median (Freq. Table)

  • Add up all the frequencies to get the total frequency. Σf
  • Divide this total by 2. This gives us the position of the median.
  • On your table, construct a cumulative frequency column. (The last value should be the same as the Σf)
    • (Cum. Freq. = add the second frequency to the first, then add the third to that total, then the forth to that total and so on)
  • Find the value corresponding to the position of the median by reading across to x

Median's position = Σf / 2 

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Mode (Freq. Table)

Choose the value with the highest frequency and read across to x.

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Range (Freq. Table)

The difference between the highest and lowest value of x.

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Finding a Missing Value from a Frequency Table when You Have Been Given the Mean

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Finding n from a Freq. Table

  • Create an fx column (and an x column if you are working with a grouped freq. table.)
  • Total all the columns.
  • Fill out the Σfx / Σf
  • Beside that put your mean / 1
  • Cross multiply
  • Finish equation
  • Your answer should always be positive and a whole number
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Selecting the Most Appropriate Average

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The Most Appropriate Average

Mean = When all the values must be taken into account.

Median = When there are one or two abnormal values.

Mode = When you want the most popular value.

Mode = When the values are not numerical.

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Modal class

The modal class is the one with the highest frequency.

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