Data Interpretation

Statistical methods

Statistical methods allow you to work on your data. Imagine you have the exam scores for 100 students, and you want to interpret this data. Scanning through the scores by eye alone might be quite tough! Here are two methods that would simplify this task.

Measures of central tendency

Central tendency values are used to describe some key characteristics of the whole data set, producing a single value that is typical of the whole set. For example, the mode will give you the value that occurs the most often.

• The mean is the most commonly reported measure of central tendency and it is the mathematical average. To calculate your mean, you add up all of your values available and divide that by the Number of values you added. The mean is represented by μ, and its formula is$\mu =\frac{\Sigma x}{n}$, where n is the Number of data items in the sample and$\Sigma x$ is the sum of all data values.

• The median is the mid-point value in your data set. Where the median is two numbers, it is the average of both values in ordered data.

• The mode is the value that occurs the most often.

Default deviation

Another statistical measure that is commonly used is variability, also known as spread. The range is the simplest form of variability. Let's take the exam score dataset again - the range is the span between the lowest and highest numerical values.

Another common measure is variance; which is the squared average deviation from the mean. This number indicates how much the individual values deviate from the mean. What you will see reported more often is the Standard Deviation. This is modelled as the square root of the variance. Standard Deviation expresses how much individual class scores differ from the mean value for the group. Mathematically, it can be modelled into an equation:

$s=\sqrt{\frac{\Sigma {(x_i-\overline{x})}^2}{n-1}}$

Single variable data

Single Variable Data involves examining one particular variable relevant to a dataset. Single data analysis is common in descriptive forms of analysis and uses Histograms, frequency Distributions, and Box Plots among other methods. This is mostly used in the first step of investigating data. Let's take a look at a box plot.

boxplot

A box plot displays a five-number summary of a dataset. They are the minimum, first quartile, median, third quartile, and maximum. Quartiles tell us about the spread of data by breaking the data set into quarters. The lower quartile, Q₁ represents 25%, the middle quartile equals 50% and the upper quartile represents 75%.

The ages of 10 students in grade 12 were collected and they are as follows:

15, 21, 19, 19, 17, 16, 17, 18, 19, 18.

Let's first arrange these in ascending order.

15, 16, 17, 17, 18, 18, 19, 19, 19, 21.

We can now find the median, which is the middle number. And since we have an even number, we have two of them. Finding the average is standard practice; however, with ours, we have the same number.

median = 18

We will find the quartiles now. The first is the median to the left of the overall median.

That will mean we are finding the median for 15, 16, 17, 17, 18.

This equals 17.

The third quartile will be the median to the right of the median.

18, 19, 19, 19, 21

Which will make that 19.

Now we will document the minimum number which is 15.

And also document the maximum which is 21.

Figure 1. Box plot

The image above is the box plot representing the data of the ages of the students in grade 12.

We will take another example with an odd number of data points.

Bivariate data

In contrast to Single Variable Data, Bivariate Data consist of two variables for each individual. For example, in large studies in the health sector, it is common to collect variables such as height, age, blood pressure, etc. in each individual. Let's look at an example in a two-way frequency table.

These are the number of males and females who had each grade on a math project in school.

We can see there are 9 males and 12 females who got an A, 18 males and 14 females who got a B, and so on.

Now we can answer a couple of questions.

1. How many students in total had an A?

Answer: 21 students.

2. How many males were surveyed?

Answer: 38 males.

3. How many males earned a grade of A?

Answer: 9.

Probability

Probability is the measure of how likely an event is to happen. Probabilities can be placed on a Number Line between 0 and 1, as shown below.

So if the Probability of an event is zero, then it is impossible for the event to occur. Whilst if it is 1, then it is certain. Then we have variant degrees in between the two values, and 0.5 would mean there is an even chance of the event happening.

Probability on a number line

Probabilities are written down using the following Notation :

• P (A): the probability of event A happening.

• P (A '): the probability of event A not happening.

If event A has a between happening and not happening, then the probability of event A not happening = 1 - P (A ')

For example, if the P (A) = 0.8

P(A') = 0.2.

They should both add up to 1.

These are the basic concepts you would be using throughout probability at this level. You can be reintroduced to Venn diagrams, tree diagrams, etc. as well!