Statistics Ch 3

Page 1

Statistics 3.1

Compute the mean of a data set

Arithmetic mean/mean- average

Mean=sum of data points/# of data points

We will round the mean to one more decimal place than the

Σ x=x1+x2+...+xn

Sample mean: mean of a sample x x̅

Population mean: mean of the population μ

Measure of center

Compute the median of a data set

Median: number that splits data set in half

Measure of center

Compare the properties of the mean and median

The mean is influenced by more extreme values than the median

Because of this the median is resistant

If the data is skewed left the median will be left of the mean, if is is symmetric it will be the same and if it is skewed right then the median will be right of the mean

Find the mode of a data set

The mode is the value that appears most frequently

If two or more are tied for most frequent, they are all considered modes

If no value appears more than once, we say that the data set has no mode

Approximate the mean with grouped data

Approximating the mean with grouped data: 1) compute the midpoint of each class. The midpoint of a class is found by taking the average of the lower class limit and the lower limit of the next larger class. For the last class, there is no next larger class, but we use the lower limit that the next larger class would have had, 2) For each class, multiply the class midpoint by the class frequency, 3) add the products Midpoint x Frequency over all classes, 4) Divide the sum obtained in Step 3 by the sum of the frequencies

Page 2

Statistics 3.2

Compute the range of a data set

The range of a data set is the difference between the largest value and the smallest value

range=largest value-smallest value

Compute the variance of a population and a sample

The variance is a measure of how far the values in a data set are from the mean, on average

x-mu (deviation)

The average of the squared deviation is the population variance

Data values from a sample use the sample variance (same formula but N-1)

Compute the standard deviation of a population and a sample

The square root of the variance is the standard deviaition

The standard deviation is not resistant

Approximate the standard deviation with grouped data

Step 1: Compute the midpoint of each class. The midpoint of a class is found by taking the average of the lower class limit and the lower class limit of the next larger class. Then compute the mean as described in Scetion 3.1

Step 2: For each class, subtract the mean from the class midpoint to obtain Midpoint-Mean

Step 3: For each class, square the difference obtained in step 2 to obtain (Midpoint-Mean)^2 and multiply by the frequency to obtain (Midpoint-Mean)^2 x frequency

Step 4: Add the products (Midpoint-Mean)^2 x frequency over all classes

Step 5: Compute the sum of the frequencies n. To compute the population variance, divide the sum obtained in step 4 by n. To compute the sample variance, divide the sum obtained in Step 4 by n-1

Step 6: Take the square root of the variance to step 5. The result is the standard deviation

Use the Empirical Rule to summarize data that are unimodel and approximately symmetric

The Empirical Rule:

68%, 95%, 99.7%

Use Chebyshev's Inequality to describe a data set

Chebyshev's Inequality:

K=2 (75%) or K=3 (88.9%)

Compute the coefficient of variation

Coefficient of variation:

CV=standard deviation/mean