This document discusses descriptive statistics and how to calculate them. It covers preparing data for analysis through coding and tabulation. It then defines four types of descriptive statistics: measures of central tendency like mean, median, and mode; measures of variability like range and standard deviation; measures of relative position like percentiles and z-scores; and measures of relationships like correlation coefficients. It provides formulas for calculating common descriptive statistics like the mean, standard deviation, and Pearson correlation.

1. DESCRIPTIVE
STATISTICS
DR. GYANENDRA NATH
TIWARI

2. TOPICS DISCUSSED IN THIS
CHAPTER
• Preparing data for analysis
• Types of descriptive statistics
– Central tendency
– Variation
– Relative position
– Relationships
• Calculating descriptive statistics

3. PREPARING DATA FOR
ANALYSIS
• Issues
– Scoring procedures
– Tabulation and coding
– Use of computers

4. SCORING PROCEDURES
• Instructions
– Standardized tests detail scoring instructions
– Teacher-made tests require the delineation of scoring
criteria and specific procedures
• Types of items
– Selected response items - easily and objectively scored
– Open-ended items - difficult to score objectively with
a single number as the result

5. TABULATION AND CODING
• Tabulation is organizing data
– Identifying all information relevant to the analysis
– Separating groups and individuals within groups
– Listing data in columns
• Coding
– Assigning names to variables
• EX1 for pretest scores
• SEX for gender
• EX2 for posttest scores

6. TABULATION AND CODING
• Reliability
– Concerns with scoring by hand and entering data
– Machine scoring
• Advantages
– Reliable scoring, tabulation, and analysis
• Disadvantages
– Use of selected response items, answering on scantrons

7. TABULATION AND CODING
• Coding
– Assigning identification numbers to subjects
– Assigning codes to the values of non-numerical or categorical
variables
• Gender: 1=Female and 2=Male
• Subjects: 1=English, 2=Math, 3=Science, etc.
• Names: 001=Rahul, 002=Rajani, 003= Rita, … 256=Harish

8. COMPUTERIZED ANALYSIS
• Need to learn how to calculate descriptive
statistics by hand
– Creates a conceptual base for understanding the
nature of each statistic
– Exemplifies the relationships among statistical
elements of various procedures
• Use of computerized software
– SPSS-Windows
– Other software packages

9. DESCRIPTIVE STATISTICS
• Purpose – to describe or summarize data in a parsimonious
manner
• Four types
– Central tendency
– Variability
– Relative position
– Relationships

10. DESCRIPTIVE STATISTICS
• Graphing data – a
frequency polygon
– Vertical axis
represents the
frequency with which
a score occurs
– Horizontal axis
represents the scores
themselves
SCORE
9.0
8.0
7.0
6.0
5.0
4.0
3.0
SCORE
Frequency
5
4
3
2
1
0
Std. Dev = 1.63
Mean = 6.0
N = 16.00

11. CENTRAL TENDENCY
• Purpose – to represent the typical score attained by subjects
• Three common measures
– Mode
– Median
– Mean

12. CENTRAL TENDENCY
• Mode
– The most frequently occurring score
– Appropriate for nominal data
• Median
– The score above and below which 50% of all scores lie
(i.e., the mid-point)
– Characteristics
• Appropriate for ordinal scales
• Doesn’t take into account the value of each and every
score in the data

13. CENTRAL TENDENCY
• Mean
– The arithmetic average of all scores
– Characteristics
• Advantageous statistical properties
• Affected by outlying scores
• Most frequently used measure of central tendency
– Formula

14. VARIABILITY
• Purpose – to measure the extent to which scores are spread
apart
• Four measures
– Range
– Quartile deviation
– Variance
– Standard deviation

15. VARIABILITY
• Range
– The difference between the highest and lowest score in a data set
– Characteristics
• Unstable measure of variability
• Rough, quick estimate

16. VARIABILITY
• Quartile deviation
– One-half the difference between the upper and lower quartiles in a
distribution
– Characteristic - appropriate when the median is being used

17. VARIABILITY
• Variance
– The average squared deviation of all scores around the mean
– Characteristics
• Many important statistical properties
• Difficult to interpret due to “squared” metric
– Formula

18. VARIABILITY
• Standard deviation
– The square root of the variance
– Characteristics
• Many important statistical properties
• Relationship to properties of the normal curve
• Easily interpreted
– Formula

19. THE NORMAL CURVE
• A bell shaped curve reflecting the distribution of many variables
of interest to educators

20. THE NORMAL CURVE
• Characteristics
– Fifty-percent of the scores fall above the mean and
fifty-percent fall below the mean
– The mean, median, and mode are the same values
– Most participants score near the mean; the further a
score is from the mean the fewer the number of
participants who attained that score
– Specific numbers or percentages of scores fall
between ±1 SD, ±2 SD, etc.

21. THE NORMAL CURVE
• Properties
– Proportions under the curve
• ±1 SD = 68%
• ±1.96 SD = 95%
• ±2.58 SD = 99%
– Cumulative proportions and percentiles

22. SKEWED DISTRIBUTIONS
• Positive – many low scores and few high
scores
• Negative – few low scores and many high
scores
• Relationships between the mean, median, and
mode
– Positively skewed – mode is lowest, median is in
the middle, and mean is highest
– Negatively skewed – mean is lowest, median is in
the middle, and mode is highest

23. MEASURES OF RELATIVE
POSITION
• Purpose – indicates where a score is in relation to all other
scores in the distribution
• Characteristics
– Clear estimates of relative positions
– Possible to compare students’ performances across two or more
different tests provided the scores are based on the same group

24. MEASURES OF RELATIVE
POSITION
• Types
– Percentile ranks – the percentage of scores that fall at or above a
given score
– Standard scores – a derived score based on how far a raw score is
from a reference point in terms of standard deviation units
• z score
• T score
• Stanine

25. MEASURES OF RELATIVE
POSITION
• z score
– The deviation of a score from the mean in standard
deviation units
– The basic standard score from which all other standard
scores are calculated
– Characteristics
• Mean = 0
• Standard deviation = 1
• Positive if the score is above the mean and negative if it is
below the mean
• Relationship with the area under the normal curve

26. MEASURES OF RELATIVE
POSITION
• z score (continued)
– Possible to calculate relative standings like the percent better than a
score, the percent falling between two scores, the percent falling
between the mean and a score, etc.
– Formula

27. MEASURES OF RELATIVE
POSITION
• T score – a transformation of a z score where T = 10(z) + 50
– Characteristics
• Mean = 50
• Standard deviation = 10
• No negative scores

28. MEASURES OF RELATIVE
POSITION
• Stanine – a transformation of a z score where the stanine = 2(z)
+ 5 rounded to the nearest whole number
– Characteristics
• Nine groups with 1 the lowest and 9 the highest
• Categorical interpretation
• Frequently used in norming tables

29. MEASURES OF
RELATIONSHIP
• Purpose – to provide an indication of the relationship
between two variables
• Characteristics of correlation coefficients
– Strength or magnitude – 0 to 1
– Direction – positive (+) or negative (-)
• Types of correlation coefficients – dependent on the
scales of measurement of the variables
– Spearman rho – ranked data
– Pearson r – interval or ratio data

30. MEASURES OF
RELATIONSHIP
• Interpretation – correlation does not mean causation
• Formula for Pearson r

31. CALCULATING DESCRIPTIVE
STATISTICS
• Symbols used in statistical analysis
• General rules for calculating by hand
– Make the columns required by the formula
– Label the sum of each column
– Write the formula
– Write the arithmetic equivalent of the problem
– Solve the arithmetic problem

32. CALCULATING DESCRIPTIVE
STATISTICS
• Using SPSS Windows
– Means, standard deviations, and standard scores
• The DESCRIPTIVE procedures
• Interpreting output
– Correlations
• The CORRELATION procedure
• Interpreting output

34. FORMULA FOR THE MEAN
n
x
X



35. FORMULA FOR VARIANCE
 
1
2
2
2





N
N
x
S
x
x

36. FORMULA FOR STANDARD
DEVIATION
 
1
2
2





N
N
x
SD
x

37. FORMULA FOR PEARSON
CORRELATION
  
   



























N
y
y
N
x
x
N
y
x
xy
r
2
2
2
2

38. FORMULA FOR Z SCORE
sx
X
x
Z
)
( 
