Processing and presenting data

In a nutshell

Raw scientific data is processed and presented in a logical manner.

Raw vs processed data

Raw data is recorded directly from the experiment, which is processed by calculations to give processed data. Anomalies are ignored when processing data. 

Example

A student investigates the incidence of heart disease in a sample size of $$500$$​ people. She records this raw data in a table:

The student wants to present the number of people with heart disease as a percentage:​

$$({88\over 500})\times 100 = 17.6~\%$$

Percentages are a form of processed data because they are calculated from the raw data.

Averages

Averages are a form of processed data and there are three types: mean, median and mode.

The mean is the sum of all values divided by the total number of values.

​$$mean = {sum~of ~all~values\over total ~number ~of ~values}$$​​

The median is the middle value.

The mode is the value that occurs most frequently in a set of data.

Another value that can be calculated from raw data is the range, which is the difference between the largest and smallest value.

​$$range = largest ~value - smallest~value$$​​

Example 

A student records how many cars that drive past her house between $$4\ pm$$​ and $$5\ pm$$​ during the week.

1. Monday - $$15$$​ cars
   
2. Tuesday - $$12$$​ cars
3. Wednesday - $$15$$​ cars
4. Thursday - $$11$$​ cars
5. Friday - $$13$$​ cars

All values in ascending order:

​

​$$11, 12, 13, 15, 15$$​​

The mode is $$\underline{15}$$​ as $$15$$​ occurs most frequently in the data.

The median is $$\underline{13}$$​ because it is the middle value.

The mean is $$\underline{13.2}$$​ but this is rounded to a whole number ($$13$$​) since there can't be $$13.2$$ cars:

$$mean ={(11+12+13+15+15)\over5} = \underline{13.2}$$

The range is $$\underline{4}$$​

$$range = 15-11 = \underline4$$​​

Significant figures 

It is important to consider the following concepts when rounding numbers according to significant figures:

PROCEDURE

Example

How many significant figures are in $$0.002034$$​?

$$0.002034$$​ has four significant figures.

Round $$0.3400606$$​ to four significant figures.

$$0.3400606$$​ rounded to four significant figures is:

​$$\underline{0.3401}$$​​

Categoric data

Data that is grouped into categories is categoric data.

Example

A student investigated the different colours of cars driving past her house. Car colour is an example of categoric data.

Continuous data 

Data that can have any numerical value. 

Example

A student investigated the height of students in her class. Height is an example of continuous data.

Calculating gradients

Gradients can be calculated from a linear graph, or from a tangent on a non-linear graph, using the following equation:

​$$gradient ={ change~in ~y\over change~in ~x }$$

​

PROCEDURE

Calculating rate

Rate of reaction is how much product is formed in a given time; it measures how quickly a product is produced and calculated by either:

​$$rate ~of~reaction = {amount~of~product~produced \over time} \\ \ \\OR \\ \ \\rate ~of~reaction={amount ~of reactant~used~up\over time}$$

In a graph where time is plotted on the $$x$$​-axis and amount of product produced or amount of reactant used up is plotted on the $$y$$-axis, the initial rate of reaction can be calculated by the gradient at $$t=0$$​. A  tangent is drawn on the graphs at $$t=0$$​ to calculate the gradient and therefore the rate:

$$rate~of~reaction = gradient={change~in~y \over~change ~in~x}$$

Example

The volume of the product produced was recorded against the time taken. Calculate the initial rate of reaction.

A. Volume ($$cm^3$$)​ B. Time ($$s$$)​ 1. Tangent for initial rate

​$$(x_1,y_1) = (0,0)\\ \ \\(x_2, y_2) = (20,70)\\ \ \\ initial~rate~of~reaction = {change~in~y\over~change~in~x}\\ \ \\ initial~rate~of~reaction=\underline{3.5 ~ cm^3~s^-1}$$​​

The initial rate of reaction is ​$$\underline{3.5 \, cm^3 s{^-1}}$$.

Correlation

Graphs can be used to illustrate the relationship between two variables. 

Graph	Type of Correlation	Explanation
	None	There is no relationship between the variable on the $$x-$$​axis and the variable on the y axis. This means there is no correlation between the two variables.
	Negative	As the variable on the $$x-$$​axis increases, the variable on the $$y-$$​axis decreases. This means there's a negative correlation between the two variables.
	Positive	As the variable on the $$x-$$​axis increases, the variable on the $$y-$$​axis also increases. This means there is a positive correlation between the two variables.