2.7 Terms

For data having a distribution that is BELL-SHAPED and SYMMETRIC:

Approximately 68% of the data is within one standard deviation of the mean. Approximately 95% of the data is within two standard deviations of the mean. More than 99% of the data is within three standard deviations of the mean. This is known as the Empirical Rule. It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.

value = mean + (#ofSTDEV)(standard deviation)

(#STDEV) does not need to be an integer.

Standard Deviation

A number that measures how far data values are from their mean. Provides a numerical measure of the overall amount of variation in a data set, and Can be used to determine whether a particular data value is close to or far from the mean.

If x is a number, then the difference "x - mean" is called its ___.

Deviation.

Variance

The average of the squares of the deviations (the x - x¯ values for a sample, or the x - μ values for a population).

The notation for the standard error of the mean is σ/√n

Where σ is the standard deviation of the population and n is the size of the sample.

#ofSTDEVs is often called a "___-_______"; we can use the symbol z.

z-score.

sample: x = x¯ + (#ofSTDEV)(s) Population: x=μ+(#ofSTDEV)(σ)

The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation. The symbol x¯ is the sample mean and the Greek symbol μ is the population mean.

If you add the deviations, the sum is always ____.

Zero.

For ANY data set, no matter what the distribution of the data is:

At least 75% of the data is within two standard deviations of the mean. At least 89% of the data is within three standard deviations of the mean. At least 95% of the data is within 4.5 standard deviations of the mean. This is known as Chebyshev's Rule.

The standard deviation is ______ when the data values are more spread out from the mean, exhibiting more variation.

Larger.

The standard error of the mean.

Measures the sampling variability of a statistic. An example of standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean.

By squaring the deviations, you make them positive numbers, and the sum will also be ____.

Positive.

How much the statistic varies from one sample to another is known as the ____ _____ of a statistic.

Sampling variability.

The standard deviation is _____ when the data are all concentrated close to the mean, exhibiting little variation or spread.

Small.