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08. Descriptive Data Analysis

The next stage of data analysis consists of descriptive and inferential data analysis. Descriptive data analysis provides the researcher a basic picture of the problem he is studying. It consists of:

  • Measures of Central Tendency
  • Measures of Dispersion
  • Measures of Skewness and Kurtosis

Measures of Central Tendency

A measure of central tendency is a value that represents a typical or central element of a data set. The important measures of central tendency are Mean, Median, and Mode.

Mean: Mean (average) is the sum of the data entries divided by the number of entries. Sample mean is denoted by X̄ and the population mean is denoted by μ.

Population Mean:

Population Mean Formula

Sample Mean:

Sample Mean Formula

Properties of Mean:

  • Data possessing an interval scale or a ratio scale, usually have a mean.
  • All the values are included in computing the mean.
  • A given set of data has a unique mean.
  • The mean is affected by unusually large or small data values (known as outliers).
  • The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

Median: The median of a data set is the middle data entry when the data set is sorted in order. If the data set contains an even number of elements, the median is the mean of the two middle entries. The median is the most appropriate measure of central tendency to use when the data under consideration are ranked data, rather than quantitative data.

Mode: The mode of a data set is the entry that occurs with the greatest frequency. A set may have no mode or may be bimodal when two entries each occur with the same greatest frequency. The mode is most useful when an important aspect of describing the data involves determining the number of times each value occurs. If the data are qualitative then mode is particularly useful.

Measurement Matrix Scale Chart

Measures of Dispersion

Measures of Dispersion indicate the amount of variation or spread, in a data set. There are four important measures of dispersion:

  • The Range: The range is the difference between the largest and smallest observation. It is sensitive to extreme values.
  • The Interquartile Range: The difference between the third and first quartiles (Q3 - Q1). It gives the middle 50% of the data.
  • Variance: The average of the squared differences between each observation and the mean.
  • Standard Deviation: The square root of the sample variance. It lends itself well to mathematical analysis.

Relationships Between Two Variables

Two of the important techniques used to study the relationship between two variables are correlation and regression.

Correlation: Measures association between two variables. In graph form it is shown as a 'scatter diagram' plotting scores on the X and Y axes. A positive relationship means values increase together. A negative relationship means as one increases, the other decreases.

Pearson's (r) and Spearman's (rho) are common correlation metrics running from -1 to +1. Interpretation criteria:

  • Equal to 0: No correlation
  • Less than 0.2: Very low
  • Between 0.2 and 0.4: Low
  • Between 0.41 and 0.70: Moderate
  • Between 0.71 and 0.90: High
  • Over 0.91: Very high
  • Equal to 1: Perfect correlation
Scatter Plot Matrix Sample

Simple Regression Analysis: Gives the equation of a straight line (y = a + bx) to allow calculation models.