Data Analysis: Descriptive Statistics

This is a presentation on descriptive statistics, which is one type of data analysis.

1. Data Analysis: Descriptive Statistics Mahmood Ahmad [email protected] 11 th February 2014

2. Synopsis Levels of Quantitative Description Types of Data Analysis Statistical Measures Applied to Descriptive Analysis Measures of central tendency/average Mean Median Mode Measure of spread/dispersion Range Variance Standard deviation Measure of relative position Standard scores Percentile rank Percentile score Measures of relationship Coefficient of correlation Computational Data Analysis

3. Scales of Quantitative Description Level Scale Process Data Some Appropriate Statistics 4 Ratio Equal intervals True zero Ratio relationship Parametric Descriptive Inferential 3 Interval Equal Intervals No true zero Mean Standard deviation Pearson’s r 2 Ordinal Ranked in order Nonparametric Median Quartile deviation Stanines Spearman’s rho ( ρ ) Mann-Whitney Wilcoxin 1 Nominal Classified and counted Mode Chi square Sign

4. Choosing Statistical Tests for Data

5. Descriptive and Inferential Analysis Descriptive Analysis Descriptive statistical analysis limits generalization to the particular group of individuals observed. That is: No conclusions are extended beyond this group. Any similarity to those outside the group cannot be assumed. The data describe one group and that group only. Examples: assessment findings, findings of a much simpler action research Inferential Analysis Inferential analysis selects a small group (sample) out of a larger group (population) and the finding are applied to the larger group. It is used to estimate a parameter , the corresponding value in the population from which the sample is selected. It is necessary to carefully select the sample or the inferences may not apply to the population.

6. Descriptive Analysis Statistical measures applied to descriptive data are as follows: Measures of central tendency/average Mean Median Mode Measure of spread/dispersion Range Variance Standard deviation Measure of relative position Standard scores Percentile rank Percentile score Measures of relationship Coefficient of correlation

7. Descriptive Analysis: Measures of Central Tendency/Average The Mean/Arithmetic Average X̄ = Median It is a measure of position in an array, above and below which one-half of the scores fall. 1 2  4 5 1 2 3 4 5 6 ( Mode It is the most frequent score in a distribution. 2 3 4 4 5 6 Some distributions may be bimodal or multimodal also.  

8. Descriptive Analysis: Measure of Spread/Dispersion Range It is difference between the highest and the lowest scores. Deviation A score expressed as its distance from the mean is called a deviation score. x=X-X̄ The deviation score may be positive (above the mean) or negative (below the mean). Variance ( σ ) It is the sum of squared deviation divided by N . Standard Deviation The square root of the variance is called the standard deviation. OR  

9. Descriptive Analysis: Measure of Relative Position/Standard Scores I. Standard Scores The Z Score (Sigma) It is the deviation from the mean in terms of standard deviation. OR T Score (T) It is derived by multiplying the z score by 10 and adding 50. OR College Board Score ( Z cb ) It is derived by multiplying the z score by 100 and adding 500. OR II. Stanines ( sta ndard+ nine ) It is a standard score that divides the normal curve into nine parts. III. Percentile Rank It is the percentage of scores in its frequency distribution that are the same or lower than it. The percentile of a data value is the percentage of scores that fall below that data. The rank is the position of the data value in an ordered list. Formula to find percentile (p) using rank (R) and total number of data entries (n ):  

10. Measures of Relationship These indicate the degree of relationship (correlation coefficient) between two or more quantifiable variables from a single group of participants . The correlation can be positive (+), negative (-), or none . When one variable increases with the other, the correlation is positive ; when one variable increases while the other decreases or the vice versa, the correlation is negative . There may be variables that have no correlation . A perfect positive correlation is +1; a perfect negative correlation is -1; a complete lack of correlation is 0. The sign of the coefficient indicates the direction of the relationship. A perfect positive correlation specifies that for every unit increase in one variable there is a proportional unit increase in the other. The perfect negative correlation specifies that for every unit increase in one variable there is a proportional unit decrease in the other.

11. Measures of Relationship: Pearson’s Product-Moment Coefficient of Correlation ( r ) It can be calculated by either of the following formulas: OR  

12. Measures of Relationship: Rank Order Correlation/Spearman’s rho ( ρ ) In this relationship, the paired variables are expressed as ordinal values (ranked) rather than as interval or ratio values. It is computed by the following formula:  

13. Computational Data Analysis Computers are used to easily and flawlessly arrange and analyze data and apply statistical formulae. The following software programs can be used for this purpose: IBM SPSS Statistics Minitab MS Excel iNZight R

14. IBM SPSS Statistics integrates with a broad range of capabilities for the entire analytical process.

15. References and Further Reading Best, J. W., & Kahn, J. V. (2006). Research in Education (10th ed.). Pearson Education Inc. Cohen, L., Manion , L., & Morrison, K. (2011). Research Methods in Education (7th ed.). Routledge . Gay, L. R., & Geoffrey E. Mills, P. A. (2011). Educational Research: Competencies for Analysis and Applications (10th ed.). Pearson.