Maths Skill: Calculating Mean Values & Standard Deviation
- Descriptive statistics are invaluable when interpreting data from experiments
- Some experiments have thousands or millions of data values/observations
- Descriptive statistics allow for sample data to be summarised in a concise manner
- Other statistics have different purposes such as:
- Testing for a significant difference between means
- Testing for correlation between variables
- Investigating discrete data (data that falls into distinct categories)
Mean
- A mean value is what is usually meant by “an average” in biology
Mean = sum of all measurements ÷ number of measurements
- Problems with the mean occur when there are one or two unusually high (or low) values in the data (outliers) which can make the mean too high (or too low) to reflect any patterns in the data
- The mean is sometimes referred to as X̄ in calculations
Standard Deviation
- The mean is a more informative statistic when it is provided alongside standard deviation
- Standard deviation measures the spread of data around the mean value
- It is very useful when comparing consistency between different data sets
- The mean must be calculated before working out the standard deviation
Worked example
15 rats were timed how long it took them to reach the end of a maze puzzle. Their times, in seconds, are given below. Find the mean time.
12, 10, 15, 14, 17,
11, 12, 13, 9, 21,
14, 20, 19, 16, 23
Step 1: Calculate the mean
12 + 10 + 15 + 14 + 17 + 11 + 12 + 13 + 9 + 21 + 14 + 20 + 19 + 16 + 23 = 226
226 ÷ 15 = 15.067
Step 2: Round to 3 significant figures
Mean (X̄) = 15.1 seconds
Worked example
The ear lengths of a population of rabbits was measured.Ear lengths (mm): 62, 60, 59, 61, 60, 58, 59, 60, 57, 56, 59, 58, 60, 59, 57Calculate the mean and standard deviation.
Step 1: Calculate the mean
Mean = 885 ÷ 15 = 59 mm
Step 2: Find the difference between each value and the mean
Subtract the mean from each value to find the difference
Example: 62 - 59 = 3
Step 3: Square each difference
Square the difference for each value
Example: 32 = 9
Step 4: Total the differences
Step 5: Divide the total by (n-1) to get value A
36 ÷ (15 - 1) = 36 ÷ 14 = 2.571
Step 6: Get the square root of value A
Standard Deviation = 1.60
Examiner Tip
Constructing a table like the one above can help you to keep track of all your calculations!Note that you won't be asked to calculate standard deviation in an exam, but it is important that you understand what it tells you about a data set. You might also need to calculate it when analysing the results of a required practical
