Analysing data

​​In a nutshell

Once an experiment is run and data has been gathered, the readings must be analysed so that an appropriate conclusion can be drawn. The mean can be calculated by summing the individual readings and dividing by how many readings there are and is used to represent the most central value of the data set. Standard form is a way of displaying very large or small numbers by giving them in relation to a power of ten. 

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Equations

Definitions

Standard form and significant figures

Standard form is a way of displaying numbers that are either very large or very small without having to write them out in full. This is achieved by writing the number in terms of a whole number between $$0$$ and $$10$$ multiplied by a power of $$10$$. Standard form is used in conjunction with significant figures, which allows numbers to be written to a specific accuracy.

The general form of a value in standard form is given as $$a \times 10^b$$, where $$a$$ is a number between $$0$$ and $$10$$ that represents the digits of the number, while $$b$$ is the power of $$10$$ that represents the size of the number and always takes the form of an integer. The value of $$b$$​ is positive when the value is large and negative when it is small.

Example

A piece of paper is measured to be $$0.0586\,mm$$ thick. Convert this to meters and write this value in standard form, to two significant figures.

First, convert millimeters into meters:

$$0.0586\,mm = 0.0000586\,m$$

Write this value to two significant figures:

$$0.0000586\,m \to 0.000059\,m$$

Finally, write this value in standard form:

$$0.000059 = 5.9\times10^{-5}\,m$$

The thickness of the piece of paper in meters is $$\underline{5.9\times10^{-5}\,m}$$.​

Calculating the mean

The mean of a set of values calculates one value that represents the central piece of data in the set. It is calculated from a set of repeat readings and is given by the following equation:

​$$mean = \dfrac{sum\ of\ results}{number\ of\ data\ points}$$​​

Note: It is important not to use anomalous results when calculating the mean, as extreme values like those may skew the value for the mean and decrease its accuracy.

Example

A measurement is repeated five times and the following readings are taken. Calculate the mean of these values to one decimal place.

First, write down the equation:

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$$mean = \dfrac{sum\ of\ results}{number\ of\ data\ points}$$​​​

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Next, substitute in the values and calculate the mean:

$$mean = \dfrac{87.2+83.4+89.8+91.4+88.8}{5}$$​​

$$mean = 88.1$$​​

The mean of these values is $$\underline{88.1}$$.

Drawing and interpreting graphs

In general, the aim of experiments is to find a relation between two variables. Drawing a graph is a good way to see this relation represented, as lines or curves can be drawn on the graph to show this. Graphs must have correctly-labelled axes, with the labels including appropriate units. 

Example

This is a graph of displacement against time. The $$y$$​-axis is labelled with $$x\,(m)$$, showing displacement measured in meters, while the $$x$$​-axis is labelled with $$t\,(s)$$, showing time measured in seconds.

Gradients and tangents

If the two chosen quantities are related in a linear way, then the graph will have a straight line of best fit through the plots. This line will follow the following relation.

​$$y = mx+c$$​​

where $$m$$ is the gradient of the line and $$c$$ is the $$y$$-intercept of the line. These values can be analysed and conclusions can be drawn, depending on the experiment and the quantities.

Sometimes a graph's relation does not fit a straight line, but rather a curved one. If a certain point needs analysing closer, it will be necessary to find the tangent of the curve at that point, depending on the quantities involved.

Non-linear graphs

​If one quantity is proportional to another quantity to some power, then the axis should be chosen accordingly so that the relation is still a straight line. This is done by plotting the graph as a function of the first quantity against the that power of the second.

The equation $$g=\dfrac{2s}{t^2}$$ can be used to obtain a value for the acceleration due to gravity by measuring the time $$t$$ that it takes an object to fall a distance $$s$$.

Plotting a graph of $$s$$ against $$t$$ would result in a graph that looks like $$y=x^2$$ as time is being squared, which does not produce a straight line.

The gradient of this graph is constantly changing, but gravity is a constant, so the curved line is not helpful.

In order to get a straight line that has a constant gradient equal to the acceleration due to gravity, the graph will have to be plotted as $$s$$ against $$t^2$$. 

Example

The line intersects with the $$y$$-axis at $$10\,m$$, and it intersects the $$x$$-axis at $$5\,s$$. Calculate the gradient.

Gradient is given by the following equation.

$$gradient = \dfrac{\Delta y}{\Delta x}$$​​

Substitute in the values and calculate the gradient:

$$gradient = \dfrac{10}{5}$$

$$gradient = 2$$​​

The gradient of the line is $$\underline{2}$$.