stat 155 introductory statistics midterm exam 1 revie · 9/19/06 midterm exam 1 review 1 stat 155...

9/19/06 Midterm Exam 1 Review 1

STAT 155 Introductory Statistics

Midterm Exam 1 Review

The UNIVERSITY of NORTH CAROLINAat CHAPEL HILL


Midterm Exam 1

• Thur. 9/21 in class• Get a couple of Scantron sheets (Students

stores )• #2 pencil • Calculator• Closed-book, closed-notes


What is Statistics for

• Statistics: the science of collecting, organizing, and interpreting data.– Data collection.– Data analysis - organize & summarize data to

bring out main features and clarify their underlying structure.

– Inference and decision theory – extract the info provided by the data and may be used as a guide for further action.


Fundamental concepts:

• Population: the entire group of individuals that we want information about.

• Sample: a part of the population that we actually examine in order to gather information.

• Sample size: number of observations/individuals in a sample.

• Statistical inference: to make an inference about a population based on the information contained in a sample.


Data contain

• Individuals: the objects described by the data;

• Variable: any characteristic of an individual. A variable can take different values for different individuals– A categorical variable places an individual into

one of several categories.– A quantitative variable takes numerical values

for which arithmetic operations such as adding and averaging make sense.


Distributions of Variables

• The distribution of a variable indicates what values the variable takes and how often it takes these values.

• For a categorical variable, distribution: categories + count/percent for each category

• For a quantitative variable, distribution: pattern of variation of its values


Examining distribution - Overall Pattern

• Shape– Graphical techniques to display distributions

• Bar graph• Pie chart• Stemplot• Histogram

– Modes: peaks of a distribution.• unimodal or multimodal

– Symmetric or skewed (to left/right)?


Center

• Mean– easy to calculate– highly affected by outliers

• Median– can be time consuming to calculate– more resistant to a few extreme observations

(sometimes outliers)– robust

• Mode, Mean and Median– relative locations for skewed/symmetric dist.– which one to use


Spread

• Standard deviation and variance– definition and calculation– sum of deviations always equals 0– square or not– why divided by (n-1)?

• Quartiles– definition and calculation– IQR– rule to identify outliers– five-number summary– boxplots– comparison with histograms and stemplots

• Range, IQR, S.D.


Deviations

• Outliers: some values that fall outside the overall pattern.

• IQR can help to identify outliers• Modified boxplots• Strategies

– Detect them, investigate their causes, correct them, or delete them, or give them individual attention.

– Use resistant methods such as median to reduce the influence of the outliers.


Linear transformation

• form:

• its effects on shape, center and spread

.bxaxnew +=


Density curves

• probability density function (pdf)• properties of pdf• comparison with histograms• mode, median, mean, quartiles and s.d. of

density curves


normal distributions

• shape– symmetric around mean;– single-peaked (unimodal);– bell-shaped.

• center and spread• the 68-95-99.7 rule (only for normal

distributions)


Standardization and z-Scores

• Effects of standardization– Standardization is a linear transformation. – Standardized values under any distribution

always have mean 0 and s.d. 1.– Effects on shape, center and spread.

• Linear transformation: normal to normal.


Standard normal distribution

• normal probability calculation• on the margin or inside the table?• pictures help

σµ−

=xz

stat 155 introductory statistics midterm exam 1 revie · 9/19/06 midterm exam 1 review 1 stat 155...

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