Data is only useful when it has meaning. It becomes significant if we compare it to similar data sets.

If we are comparing Grade 12 students sitting an AP Mathematics exam, we may be interested in which class has the highest mark. Perhaps the class is filled with geniuses, or maybe the class has a good teacher. We would also like to know the spread of the exam results.

Do all the students in a class have a similar ability or is the class mixed-ability? Perhaps the school determines classes on sets.

How do we determine if a student should be in that set? Here we are looking at extreme values (outliers).

If a student is way beyond the ability of the class, he will need to move up. This is an example of analyzing one-variable data.

Discrete or Continuous Data

Data can be divided into two types: discrete or continuous.

For example, the number of children in a household is considered discrete data, since the number can only be an integer. Another example would be the height to the nearest meter, as we do not accept decimals, and hence can only be an integer.

When representing discrete data, we tabulate the frequency for specific values.

In contrast to discrete data, there's also:

Measures such as weight, or time to complete a lap are examples of continuous data.

For this type of data, you record the frequency for intervals.

Measures of Central Tendency

The measures of central tendency reveal what is widespread or characteristic about a variable. The mode, median, and mean are the three measurements of central tendency. These averages summarize the data. It is important to know when these measures are most useful and also be aware of their disadvantages.

Mean

The mean is prevalently used in measuring the central points of data. 

Though commonly used for continuous data, it is equally perfect for discrete data.

You will often observe that the computed value of the mean is usually not represented in a given data set. Nevertheless, it reduces the tendency to misrepresent any one observation in the data set. This is because it includes all observations in determining its value.

The major problem with the mean is that it considers extreme values –the outliers —, which could be erroneous. The mean may therefore not represent the majority of the data.

Median

When determining the median, it is important to take into account the total number of data points in a data set:

• when the data set is even, you take the middle two values, add them, and divide the result by 2 to obtain the median;

• when the data set is odd, you take the value in the middle as the median itself.

There is little or no impact of outliers in this measure.

Mode

Here's an example.

The mode is often used for categorical variables and rarely used for continuous variables. When the most occurring data point is also farther away from the rest of the data set – it's an outlier –, the mode does not give a good measure.

When a data set also has two modal points (bimodal), it is inappropriate to use mode as a measure of central tendency. 

Measures of Spread

Variance, though it measures spread, is quite a poor metric of spread but significant in determining the standard deviation of a distribution; which gives a better measure.

For example, if the variance of a distribution is 32, 000, it only indicates that it is large but tells us nothing more about the distribution. Taking the square root of the variance gives the standard deviation which explains the margin of deviation of each data point from the mean value of a data set.

The variance of a sample is computed using the formula;

\(s^{2}= {Σ (x_{i}-x̄)^2\over n-1}\)

where \(x_{i}\) represents each data point and \(\bar{x}\) is the computed mean.

n is the sample size, and ∑ is summation notation. You need to divide by n-1 because we are using a sample rather than the entire population. This sample will not consider all the extreme values in the population so in order to be a better estimate of the population variance you need to increase the value of the estimate by dividing by n -1.

Given X = (12, 13, 24, 24, 25, 34). Find the sample variance and standard deviation.

Solution:

\(s^{2}= {Σ (x_{i}-x̄)^2\over n-1}\)

First, find the mean:

\bar{x} = \({ 12+13+24+24+25+34 \over 6 } = 22 \)

Second is to find variance (subtract the mean from each observation, squared and divide by the degrees of freedom): 

\(s^{2}= {Σ (x_{i}-x̄)^2\over n-1}\)

\(s^{2}= {(12-22)^2+(13-22)^2+(24-22)^2+(24-22)^2+(25-22)^2+(34-22)^2\over 5}\)

\(s^{2}= {100+81+4+4+9+144\over 5} = 68.4\)

Third, is to find the standard deviation by taking the square root of variance:

Therefore, standard deviation is approximately 8.27.

Position of a Term in a Distribution