Statistical Averages and Spreads

3 types

Mean = Total of all values / Number of values

Median = The middle value of an ordered list

Mode = Value that occurs most frequently 

Range = Highest value - Lowest value

Comparing The Different Averages

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

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.

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.

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

Finding Averages and the Spread from a Frequency 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

• 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 

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

The difference between the highest and lowest value of x.

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

• 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

Selecting 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.

The modal class is the one with the highest frequency.