The document by Hiba Armouche provides an overview of descriptive statistics, distinguishing between statistics (characteristics of samples) and parameters (characteristics of populations). It covers various types of numerical data, scores, and techniques for summarizing quantitative data, including frequency distributions, histograms, and measures of central tendency like mean, median, and mode. Additionally, it discusses variability and relative standing through concepts such as standard deviation, percentiles, and skewness in data distributions.

1. Descriptive Statistics
Presented by
Hiba Armouche

2. Outline
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Statistics versus Parameters
Types of Numerical Data.
Types of Scores
Techniques for Summarizing
Quantitative Data

3. Statistics Versus Parameters
• A parameter is a characteristic of a
population. It is a numerical or graphic
way to summarize data obtained from
the population
• A statistic is a characteristic of a
sample. It is a numerical or graphic way
to summarize data obtained from a
sample

4. Types of Numerical Data
• There are two types of data :
1- Quantitative data are obtained by determining
placement on a scale that indicates amount or degree
Ex:The temperatures recorded each day during the
months of September through December in Lebanon in
a given year (the variable is temperature )
2- Categorical data are obtained by determining the
frequency of occurrences in each of several categories
Ex: The number of male and female students in a
chemistry class (the variable is gender )

5. Types of Scores
Raw Score is the initial score obtained
Ex: The number of items an individual gets correct
on a test.
Derived Score is obtained by taking the raw score and
converting it into a more useful score

6. Types of Scores

7. Types of Scores
Age and Grade level Equivalent
tell us of what age or grade an
individual score is typical.

8. Types of Scores
A percentile rank refers to the
percentage of individuals scoring at or
below a given raw score.
PR =
Number of Students
Below Score
+
All Students
All Scores
Total Number in the Group
X 100

9. Types of Scores
Standard scores
indicate how far a given raw score
is from a reference point.
The z scores and the t scores

10. Techniques for Summarizing
Quantitative Data
• A frequency distribution is two-column
listing, from high to low, of all the scores
along with their frequencies

11. Techniques for Summarizing
Quantitative Data
• A frequency polygon is a graphic display of
frequency distribution. It is a graphic way to
summarize quantitative data for one variable
– A graphic distribution of scores in which only a few
individuals receive high scores is called a
positively skewed polygon
– One in which only a few individuals receive low
scores is called a negatively skewed polygon

12. Techniques for Summarizing
Quantitative Data
• A histogram is a bar graph used to
display quantitative data at the interval
or ratio level of measurement

13. Techniques for Summarizing
Quantitative Data
• The stem-leaf plot is a display that organizes a set of
data to show both its shape and distribution. Each
data value is split into a stem and a leaf.
The leaf is the last digit of a number.
The other digits to the left of the leaf form the stem
Example
Stem
15 9
Leaf

14. Techniques for Summarizing
Quantitative Data
• The normal distribution is a theoretical
distribution that is symmetrical and in which a
large proportion is concentrated in the middle
• The distribution curve of a normal distribution
is called a normal curve. It is a bell-shaped,
and its mean, mode, and median are identical

15. How do you analyze the data?
Conduct descriptive analysis
Descriptive Statistics
Central
Tendency
Mean
Median
Mode
Variability
Relative Standing
Variance
Standard Deviation
Range
Z-Score
Percentile Ranks

16. Averages/Measures of
central tendency
• Mode:
– The most frequently occurring score
– Appropriate for nominal data

17. Averages/Measures of
central tendency
• Median
– The score above and below which
50% of all scores lie (i.e., the midpoint)
– Characteristics
• Appropriate for ordinal scales
• Doesn’t take into account the value
of each and every score in the data

18. Averages/Measures of
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

19. 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

20. Variability or Spreads
• Purpose – to measure the extent to
which scores are spread apart
Distribution A: 19, 20, 25, 32, 39
Distribution B: 2, 3, 25, 30, 75

21. Variability or Spreads
– Range
– Quartile deviation
– Boxplots
– Variance & Standard deviation

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

23. Variability or Spreads
Quartiles and the Five-Number Summary
A percentile in a set of numbers is a value below
which a certain percentage of numbers fall and above
which the rest of the numbers fall.
Example: You received in SAT score
“Raw score 630, percentile 84”
This means that your score is 630 and 84% of those
who took the exam scored lower than you.

24. Variability or Spreads
Quartiles and the Five-Number Summary
NB:
• The median is the 50th percentile
• The first quartile is the 25th percentile Q1
• The third quartile is the 75th percentile Q3.

25. Variability or Spreads
Five-Number Summary
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•
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The lowest score
Q1
The highest score
The median
Q3
Interquartile range
IQR = Q3 - Q1

26. Variability or Spreads
• Boxplots

27. Variability or Spreads
• Standard Deviation SD
It is a single number that represents
the spread of a distribution. Every score
in the distribution is used to calculate it.

28. Variability or Spreads
• How to calculate the Standard Deviation
1- Calculate the mean
2- Subtract the mean from each score
3-Square each of these scores
4- Add all the squares of these scores
5- Divide the total by the total numbers of scores
The result is called Variance.
6- Take the square root of the variance.
This is the standard deviation

29. Variability or Spreads
SD =

30. Variability or Spreads
NB:
The more spread out scores are the
greater the deviation scores will be and
hence the larger the standard deviation

31. Relative Standing
• 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

32. Thank You
hiba.armouche@yahoo.com
www.facebook.com/TrainerHibaArmouche