Chapter 3 math

Empirical) Rule

Theorem (Math Fact): If a histogram of the data is bell shaped

Probability theory is complex and deep enough

to warrant (at least) its own course (taught in most mathematics or statistics departments)

An outlier is an

unusually small or unusually large value in a data set.

Chebyshev's Theorem At least 75% of the data values must be within

z = 2 standard deviations of the mean.

Chebyshev's Theorem At least 89% of the data values must be within

z = 3 standard deviations of the mean.

Chebyshev's Theorem At least 94% of the data values must be within

z = 4 standard deviations of the mean

For symmetric graphs, skewness is

zero and the mean is equal to the median

Probability values are always assigned on a scale from

0 to 1

The closer the correlation is to zero,

the weaker the relationship

Range is very sensitive

to the smallest and largest data values

The coefficient can take on values between

-1 and +1

The correlation coefficient ranges from which two values?

-1 and +1

Five summary

1. smallest value 2.First quartile (Q1) 3. Median (Q2) 4. Third quartile (Q3) 5. Largest value

Theorem (Math Fact): If a histogram of the data is bell shaped, as shown in the figure to the right, then all of the following are true: (one)

About 68% of all observations will fall within one standard deviation of the mean (that is, within 𝑥ҧ ± 1𝑠) About 95% of all observations will fall within two standard deviations of the mean (that is, within 𝑥ҧ ± 2𝑠) About 99.7% of all observations (almost the entire data set) will fall within three standard deviations of the mean (that is, within 𝑥ҧ ± 3𝑠)

Chebyshev's Theorem

At least 75% of the data values must be within z = 2 standard deviations of the mean At least 89% of the data values must be within z = 3 standard deviations of the mean. At least 94% of the data values must be within z= 4 standard deviations of the mean

_____ can be used to make statements about the proportion of data values that must be within a specified number of standard deviations of the mean, regardless of the shape of the distribution.

Chebyshev's Theorem

Chebyshev's Theorem:

For a distribution of any shape (symmetric, skewed, bimodal, etc.), the proportion of observations that that lie within 𝑘 standard deviations of the mean is at least: 1 − 1 , for 𝑘 > 1

interquartile range

IQR = Q3 - Q1

_____ can be used to determine the percentage of data values that must be within one, two, and three standard deviations of the mean for data having a bell-shaped distribution.

The empirical rule

For Chebyshev's theorem to hold, what should be the value of z?

Z should be greater than one

A box plot is a graphical representation of data that is based on _____.

a five-number summary

The covariance is

a measure of the linear association between two variables.

Positive values of covariance indicate _____.

a positive relation between the x and y variables

A probability near one indicates

an event is almost certain to occur.

A probability near zero indicates

an event is quite unlikely to occur

Two descriptive measures of the relationship between two variables are

covariance and correlation coefficient

A box plot is a

graphical summary of data that is based on a five- number summary.

When the data is positively skewed, what is the relationship between the mean and the median? If the data is negatively skewed, is skewness positive, zero or negative?

he mean is greater than the median. The data is negatively skewed

Box plots provide another way to

identify outliers

Correlation

is a measure of linear association and not necessarily causation. 𝑟 =𝑠𝑥𝑦/S𝑥S𝑦

Probability

is a numerical measure of the likelihood that an event will occur.

The interquartile range of a data set

is the difference between the third quartile and the first quartile.

Probability theory

is the mathematical framework by which we can make scientifically sound claims about a process (or population) using a sample

Correlation Coefficient Just because two variables are highly correlated,

it does not mean that one variable is the cause of the other.

The coefficient of variation indicates how large the standard deviation is relative to the _____.

mean

For data skewed to the left, the skewness is _____

negative

Negative values indicate a

negative relationship

The range

of a data set is the difference between the largest and smallest data values. Range = Largest value - Smallest value

The covariance Positive values indicate a

positive relationship

range is the

simplest measure of variability.

An important measure of the shape of a distribution is called

skewness

If graph is moderately skewed left then

skewness is negative and mean will usually be less than the median

If graph is moderately skewed to the right,

skewness is positive and the mean will usually be greater than the median

The variance is equal to the _____

squared value of the standard deviation

Variance is in terms of squared units, which is hard to interpret. To get back to the original units we take the (positive) square root of 𝑠2. This gives the

standard deviation

Correlation Coefficient Values near -1 indicate a

strong negative linear relationship

Correlation Coefficient values near +1 indicate a

strong positive linear relationship

sample variance

s²

A data value with a z-score less

than -3 or greater than +3 might be considered an outlier.

A key to the development of a box plot is

the computation of the median and the quartiles Q1 and Q3.

interquartile range is for

the middle 50% of the data.

interquartile range overcomes

the sensitivity to extreme data values

The z-score is often called

the standardized value.