Mean is affected by extreme values (outliers)

is the “middle” number (50% above, 50% below)

Median is not affected by extreme values

The location of the median when the values are in numerical order (smallest to largest)

If the number of values is odd, the median is the middle number

If the number of values is even, the median is the average of the two middle numbers

Mode is not affected by extreme outliers

middle value of ranked data

most frequent value

is generally used, unless extreme values (outliers) exist.

is often used, since the it is not sensitive to extreme values.

In some situations, it makes sense to report both the _____ and the ______.

ratio interval

Ratio Interval Ordinal

Ratio Interval Ordinal Nominal

numerical center of the data

Sum of deviations from the mean is zero

Computed only from the center values

Sensitive to extreme values

Not sensitive to extreme values

Does not use information from all the data

May not reflect the center

May not exist

Might have multiple modes

give information on the spread or variability or dispersion of the data values.

Describe the picture

Simplest measure of variation

No causal effect is implied

Difference between the largest and the smallest values

Only concerned with the strength of the relationship

How to compute range?

Why the range can be misleading?

average (approximately) of squared deviations of values from the mean

Formula of sample variance

Formula of sample covariance

Formula of z-score

most commonly used measure of variation

shows variation about the mean

is the square root of the variance

has the same units as the original data

Formula of sample standard deviation

Describe the image

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Can be used to compare the variablity of two or more sets of data measured in different units

Formula of coefficients of variation

To compute the _____ of a data value, subtract the mean and divide

by the standard deviation.

measures the strength of the linear relationship between

two numerical variables (X & Y)

A data value is considered an extreme outlier if its Z-score is less

than _____ or greater than _____.

The larger the absolute value of the Z-score, the farther the data

value is from the mean.

Suppose the mean math SAT score is 490, with a standard deviation of 100. Compute the Z-score for a test score of 620.

Measures the amount of asymmetry in a distribution

A tool that is helpful in describing data in certain circumstances is

called the _____

is approximates the variation of data in a bell-shaped distribution

In Empirical Rule, approximately _____ of the data in a bell shaped distribution is within ± one

standard deviation of the mean or

Measures the relative concentration of values in the center of

a distribution as compared with the tails

split the ranked data into 4 segments with an equal

number of values per segment

Describe the shape of Skewness

Describe the shape of Skewness

Describe the shape of Skewness

Describe the shape of Kurtosis

Describe the shape of Kurtosis

Describe the shape of Kurtosis

Formula of Weighted Mean for a Population

Formula of Weighted Mean for a Sample

Formula of Sample Coefficient of Correlation

Formula of Percentile

Formula of Quartiles

Myers & Associates Recently, the law firm of Myers & Associates was involved in litigating a discrimination suit concerning ski instructors at a ski resort in Colorado. One ski instructor from Germany had sued the operator of the ski resort, claiming he had not received equitable pay compared with the other ski instructors from Norway and the United States. In preparing a defense, the Myers attorneys planned to compute the mean annual income for all seven Norwegian ski instructors at the resort. However, because these instructors worked different numbers of days during the ski season, a weighted mean needed to be computed. The following data and weights were determined:

The Henson Trucking Company is a small company in the business of moving people

from one home to another within the Dallas, Texas, area. Historically, the owners have

charged the customers on an hourly basis, regardless of the distance of the move within

the Dallas city limits. However, they are now considering adding a surcharge for moves

over a certain distance. They have decided to base this charge on the 80th percentile.

They have a sample of travel-distance data for 30 moves. These data are as follows:

The first quartile, Q1, is the value for which 25% of the observations

are smaller and 75% are larger

Q2 is the same as the median (50% of the observations are smaller

and 50% are larger)

Only 25% of the observations are greater than the third quartile

is Q3 – Q1 and measures the spread in the middle 50% of the

data

is also called the midspread because it covers the middle 50%

of the data

is a measure of variability that is not influenced by outliers or

extreme values

A Graphical display of the data based on the five-number summary

can be shown in either a vertical or horizontal orientation

Measures like Q1, Q3, and IQR that are not influenced by outliers are

called

Solving for IQR: Any number greater than this is an outlier

Solving for IQR: Any number less than this is an outlier.

The five numbers that help describe the center,

spread and shape of data are:

Relationships among the five-number summary and

distribution shape

Median – Xsmallest > Xlargest – Median

Q1 – Xsmallest > Xlargest – Q3

Median – Q1 >Q3 – Median

Relationships among the five-number summary and

distribution shape

Median – Xsmallest ≈ Xlargest – Median

Q1 – Xsmallest ≈ Xlargest – Q3

Median – Q1 ≈ Q3 – Median

Relationships among the five-number summary and

distribution shape

Median – Xsmallest < Xlargest – Median

Q1 – Xsmallest < Xlargest – Q3

Median – Q1 < Q3 – Median

Shape of Boxplots: If data are symmetric around the median then the box and central line are

centered between the endpoints

Determine the shape of distribution and the box plot

Determine the shape of distribution and the box plot

Determine the shape of distribution and the box plot

A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature

24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

Find the mean and the five-number summary

The following list displays the scores of the latest test in Ben Faire’s algebra class.

83 76 82 62 57 82 83 72 76 74 90 84. Find the standard deviation and variance.

Grades on a history exam follow a normal distribution with a mean of 78 and a standard deviation 6. Find the range around the mean that includes 95% of the grades. Use the Empirical Rule formula

The heights of women follow a bell-shaped distribution with a mean of 160 cm and a standard deviation of 7.5 cm. What is the approximate percentage of women between 137.5 cm and 182.5 cm. Use the Empirical Rule formula.

Regardless of how the data are distributed, at least (1 - 1/k2) x

100% of the values will fall within k standard deviations of the

mean (for k > 1)

What proportion of the data is within 2 standard deviation of the mean?

The average price of a new car is $36,000, with a standard deviation of $4,100. What is the main percentage of card that should sell between $22,000 and $50,000?

It is not possible to determine the relative strength of the relationship

from the size of the covariance

X and Y tend to move in the same direction

X and Y tend to move in opposite directions

X and Y are independent

Measures the relative strength of the linear relationship between two

numerical variables

The closer to –1, the weaker the negative linear relationship

The closer to 1, the stronger the positive linear relationship

The closer to 0, the stronger the linear relationship

Data analysis is objective

Should report the summary measures that best

describe and communicate the important aspects of

the data set

Data interpretation is subjective

Should be done in fair, neutral and clear manner