descriptive statistics

Presented by

Hiba Armouche

Descriptive Statistics

Outline

• Statistics versus Parameters

• Types of Numerical Data.

• Types of Scores

• Techniques for Summarizing Quantitative Data

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

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 )

Types of Scores

Raw Score is the initial score obtained

Derived Score is obtained by taking the raw score andconverting it into a more useful score

Ex: The number of items an individual gets correct on a test.

Types of Scores

Types of Scores

Age and Grade level Equivalent

tell us of what age or grade an

individual score is typical.

Types of Scores

A percentile rank refers to the

percentage of individuals scoring at or

below a given raw score.

PR =

Number of Students All StudentsBelow Score + All Scores

Total Number in the Group

X 100

Standard scoresindicate how far a given raw score is from a reference point.The z scores and the t scores

Types of Scores

Techniques for Summarizing Quantitative Data

• A frequency distribution is two-column listing, from high to low, of all the scores along with their frequencies

• 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



• A histogram is a bar graph used to display quantitative data at the interval or ratio level of measurement


• 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

159

LeafStem

• 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


How do you analyze the data? Conduct descriptive analysis

Descriptive Statistics

Central Tendency

Variability Relative Standing

MeanMedianMode

VarianceStandard Deviation

Range

Z-ScorePercentile Ranks

Averages/Measures of 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


• Mean– The arithmetic average of all scores– Characteristics

• Advantageous statistical properties• Affected by outlying scores• Most frequently used measure of

central tendency– Formula

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

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

– Range– Quartile deviation– Boxplots– Variance & Standard deviation


Variability or Spreads• Range

– The difference between the highest and lowest score in a data set

– Characteristics• Unstable measure of variability• Rough, quick estimate

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.


NB:• The median is the 50th percentile• The first quartile is the 25th percentile Q1• The third quartile is the 75th percentile Q3.

Quartiles and the Five-Number Summary


Five-Number Summary• The lowest score• Q1• The highest score• The median• Q3

Interquartile range

IQR = Q3 - Q1


• Boxplots


• 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.

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


SD =

Variability or SpreadsNB:

The more spread out scores are the

greater the deviation scores will be and

hence the larger the standard deviation

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

Thank You

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descriptive statistics

Education

quantitative data

types of data

score raw score

types of scores raw

low scores

set of data

data value

graphic distribution