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

Techniques for Summarizing Quantitative Data

Techniques for Summarizing Quantitative Data

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

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

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

Techniques for Summarizing Quantitative Data

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

Averages/Measures of central tendency

• 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

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

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

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.

Variability or Spreads

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

Variability or Spreads

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

Interquartile range

IQR = Q3 - Q1

Variability or Spreads

• Boxplots

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.

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

Variability or Spreads

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

[email protected]/TrainerHibaArmouche