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