B AD 6243: Applied Univariate Statistics Understanding Data and Data Distributions Professor Laku Chidambaram Price College of Business University of Oklahoma

B AD 6243: Applied Univariate Statistics

Understanding Data and

Data Distributions

Professor Laku Chidambaram

Price College of Business

University of Oklahoma

Summarizing Data

BAD 6243: Applied Univariate Statistics 3

Measures of Central Tendency• Mean

– Summarizing sample data (continuous data)– Estimating the population parameter (µ) from the sample

statistic (x)– Arithmetic average (sum of scores/number of scores)

• Median– Mid point of distribution– Can be used in summarizing ordinal data

• Mode– Most frequently occurring value– Does not consider distribution of all scores– E.g., more males than females


• Take a sample distribution of hourly billing rates for a group of IT workers:30, 20, 40, 50, 60, 60, 100, 60, 30

x = 450

n = 9

Mean = 50

Median: 20, 30, 30, 40, 50, 60, 60, 60, 100 = 50

Mode = 60

An Example


Measures of Dispersion

• Range– Difference between highest and lowest score– Does not take into account all scores in distribution

• Variance– Measure of how much scores deviate from mean on

average– Use sample statistic (s2) to estimate population

parameter (2)• Standard Deviation

– Square root of the variance– Measure of consistency of scores


Example (contd.)

x x-x (x-x)2

20 -30 90030 -20 40030 -20 40040 -10 10050 0 060 10 10060 10 10060 10 100

100 50 250050 0 4600

Average Sum Sum

Range = 80

(x-x)2 = 4600 (sum of squared differences)

s2 = (x-x)2/n-1 = 4600/8 = 575 (variance)

s = 24 (standard deviation)


Frequencies

Box Plot



Nature of the Distribution• Skewness

– Symmetry of the distribution– Value is zero in normal distribution– If skewed positively (pile up of scores on left) or negatively (pile up

of scores on right), standardized z scores can be useful

• Kurtosis– “Peak”edness of the distribution– Value is zero in normal distribution– If positive (peaked) or negative (flat), standardized z scores can be

useful

• Statistical TestsTests of Normality

.227 9 .199 .909 9 .310VAR00001Statistic df Sig. Statistic df Sig.

Kolmogorov-Smirnova

Shapiro-Wilk

Lilliefors Significance Correctiona.


A Summary of Results


Histogram with Normal Curve

Understanding Data Distributions


Data Distributions• A data distribution is a way of representing the

frequency of occurrence of values for a variable• Data distributions can be discrete (e.g.,

Bernoulli, Binomial, Poisson) or continuous (e.g., Normal, Exponential, Uniform)

• A histogram, representing a probability density function, depicts a data distribution

• Data distributions are defined by the functional form and the values of parameters

• Our focus is on the shape of such distributions and their implications for statistical inference


Normal Distribution• Refers to a family of distributions (a.k.a

Gaussian distributions) that are bell shaped and:– Represent a continuous probability distribution– Are symmetric (with scores concentrated in the

middle)– Can be specified mathematically in terms of two

parameters: the mean (μ) and the standard deviation (σ)

– Have one mode– Are asymptotic


An Example


Standard Normal Distribution

The area P under the standard normal probability curve, with the respective z-statistic


The z Distribution• The standard normal distribution, sometimes called the z

distribution (as indicated by the formula below), is a normal distribution with a mean of 0 and a standard deviation of 1

• Normal distributions can be transformed to a standard normal distribution using the formula:

where X is a score from the original normal distribution, μ is its mean and σ is the standard deviation

• A z-score represents the number of standard deviations above or below the mean

• Note that the z distribution will only be a normal distribution if the original distribution (X) is normal


Areas Under the Curve

The Empirical Rule: 68-95-99.7


An Example

• If IQ scores are normally distributed, with a mean of 100 and a standard deviation of 15, – what proportion of scores would be greater

than 125?– what proportion of scores would fall between

90 and 120?– what proportion of scores would be less than

85?


Some Key Concepts

• Central Limit Theorem– As sample size increases, the sampling distribution of

the mean for simple random samples of n cases, taken from a population with a mean equal to and a finite variance equal to 2, approximates a normal distribution

• Sampling Distribution of the Mean*• Standard Deviation vs. Standard Error of

the Mean• Sample Size vs. Number of Samples• Other Distributions

The Central Limit Theorem

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B AD 6243: Applied Univariate Statistics Understanding Data and Data Distributions Professor Laku Chidambaram Price College of Business University of Oklahoma