1.1.6 Mathematical Analysis of Results

• Quantitative investigations of variation can involve the interpretation of mean values and their standard deviations
• A mean value describes the average value of a data set
• Standard deviation is a measure of the spread or dispersion of data around the mean
• A small standard deviation indicates that the results lie close to the mean (less variation)
• Large standard deviation indicates that the results are more spread out

Two graphs showing the distribution of values when the mean is the same but one has a large standard deviation and the other a small standard deviation

Comparison between groups

• When comparing the results from different groups or samples, using a measure of central tendency, such as the mean, can be quite misleading
• For example, looking at the two groups below
  • Group A: 2, 15, 14, 15, 16, 15, 14
  • Group B: 1, 3, 10, 15, 20, 22, 20

• Both groups have the same mean of 13
• However, most of the values in group A lie close to the mean, whereas in group B most values lie quite far from the mean
• For comparison between groups or samples it is better practice to use standard deviation in conjunction with the mean
• Whether or not the standard deviations of different data sets overlap can provide a lot of information:
  • If there is an overlap between the standard deviations then it can be said that the results are not significantly different
  • If there is no overlap between the standard deviations then it can be said that the results are significantly different 

Step one: find the full range of values included within the standard deviations for each data set

Experimental group before: 0.67 to 0.71mm

                                                              Experimental group after: 0.71 to 0.77mm

Control group before: 0.69 to 0.73mm

                                                                  Control group after: 0.67 to 0.77mm

Step two: use this information to form your answer

There is an overlap of standard deviations in the experimental group before and after the experiment (0.67~0.71mm and 0.71~0.77mm) so it can be said that the difference before and after the experiment is not significant; [1 mark]

 There is also an overlap of standard deviations between the experimental and control groups after the eight weeks (0.71~0.77mm and 0.67~0.77mm) so it can be said that the difference between groups is not significant; [1 mark]

• Plotting data from investigations in the appropriate format allows you to more clearly see the relationship between two variables
• This makes the results of experiments much easier to interpret
• First, you need to consider what type of data you have:
  • Qualitative data (non-numerical data e.g. blood group)
  • Discrete data (numerical data that can only take certain values in a range e.g. shoe size)
  • Continuous data (numerical data that can take any value in a range e.g. height or weight)

• For qualitative and discrete data, bar charts or pie charts are most suitable
• For continuous data, line graphs or scatter graphs are most suitable
  • Scatter graphs are especially useful for showing how two variables are correlated (related to one another)

Tips for plotting data

• Whatever type of graph you use, remember the following:
  • The data should be plotted with the independent variable on the x-axis and the dependent variable on the y-axis
  • Plot data points accurately
  • Use appropriate linear scales on axes
  • Choose scales that enable all data points to be plotted within the graph area
  • Label axes, with units included
  • Make graphs that fill the space the exam paper gives you
  • Draw a line of best fit. This may be straight or curved depending on the trend shown by the data. If the line of best fit is a curve make sure it is drawn smoothly. A line of best-fit should have a balance of data points above and below the line
  • In some cases, the line or curve of best fit should be drawn through the origin (but only if the data and trend allow it)

Using a tangent to find the initial rate of a reaction

• For linear graphs (i.e. graphs with a straight-line), the gradient is the same throughout
  • This makes it easy to calculate the rate of change (rate of change = change ÷ time)

• However, many enzyme rate experiments produce non-linear graphs (i.e. graphs with a curved line), meaning they have an ever-changing gradient
  • They are shaped this way because the reaction rate is changing over time

• In these cases, a tangent can be used to find the reaction rate at any one point on the graph:
  • A tangent is a straight line that is drawn so it just touches the curve at a single point
  • The slope of this tangent matches the slope of the curve at just that point
  • You then simply find the gradient of the straight line (tangent) you have drawn

• The initial rate of reaction is the rate of reaction at the start of the reaction (i.e. where time = 0)

Step 1: Estimate the extrapolated curve of the graph

Step 2: Find the tangent to the curve at 0 seconds (the start of the reaction)

The tangent drawn in the graph above shows that 72 cm³ of product was produced in the first  20 seconds.

Step 3: Calculate the gradient of the tangent (this will give you the initial rate of reaction):

Gradient = change in y-axis ÷ change in x-axis

Initial rate of reaction = 72 cm³ ÷ 20 s

Initial rate of reaction = 3.6 cm³ s^-1