### 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**

- If there is an

#### Worked Example

A group of scientists wanted to investigate the effects of a specific diet on the risk of coronary heart disease. One group was given a specific diet for 8 weeks, while the other group acted as a control. After the 8 weeks scientists measured the diameter of the lumen of the main artery in the arm of the volunteers. The results of the experiment are shown in Table 1 below:

Use the standard deviations above to evaluate whether the diet had a significant effect?** **

**[2 marks]**

**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]

#### Exam Tip

The standard deviations of a data set are not always presented in a table, they can also be represented by standard deviation **error bars** on a graph.

### Plotting & Interpreting Graphs

- 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)

- Scatter graphs are especially useful for showing how two variables are

#### 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)

- The data should be plotted with the

#### 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**

- They are shaped this way because the
- 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

- A tangent is a
- The
**initial**rate of reaction is the rate of reaction at the**start**of the reaction (i.e. where**time = 0**)

#### Worked Example

The graph below shows the results of an enzyme rate reaction. Using this graph, calculate the initial rate of reaction.

**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 cm ^{3}** 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 cm^{3} ÷ 20 s

Initial rate of reaction = **3.6 cm ^{3} s^{-1}**

#### Exam Tip

When drawing tangents: always use a ruler and a pencil; make sure the line you draw is perfectly straight; choose the point where the tangent is to be taken and slowly line the ruler up to that point; try to place your ruler so that none of the line of the curve is covered by the ruler (it is much easier if the curve is entirely visible whilst the tangent is drawn).

There is a handy phrase to help you remember how to calculate the gradient of a tangent or line. **Rise over run **means that any increase/decrease vertically should be divided by any increase/decrease horizontally.