3-2 Measures of Center

Intermediate Values

When applying rounding rules, round only the final answer, not __________ that occur during calculations. For example, the mean of 2, 3, and 5 is 3.333333 ..., which is rounded to 3.3, which has one more decimal place than the original values of 2, 3, and 5. As another example, the mean of 80.4 and 80.6 is 80.50 (one more decimal place than was used for the original values).

One Mode, More Than One Mode, or No Mode

When finding the mode, a data set can have __________, __________, or __________.

Measure of Center

A __________ is a value at the center or middle of a data set.

Critical Thinking

Although we can calculate measures of center for a set of sample data, we should use __________ to determine whether the results are reasonable. Previously, we noted that it does not make sense to do numerical calculations with data at the nominal level of measurement, because those data consist of names, labels, or categories only, so statistics such as the mean and median are meaningless. We should also think about the sampling method used to collect the data. If the sampling method is not sound, the statistics we obtain may be very misleading.

Original Values

Because the mode is one or more of the __________, we do not round values of the mode; we simply use the same data.

Midrange

The __________ of a data set is the measure of center that is the value midway between the maximum and minimum values in the original data set. It is found by adding the maximum data value to the minimum data value and then dividing the sum by 2.

Multimodal

When more than two data values occur with the same greatest frequency, each is a mode and the data set is said to be __________.

No Mode

When no data value is repeated, we say that there is __________.

Median

Important properties of the __________ are: • It does not change by large amounts when we include just a few extreme values (so it is a resistant measure of center). • It does not use every data value.

Mean

Important properties of the __________ include: • Samples drawn from the same population tend to vary less than other measures of center. • Uses every data value. • A disadvantage is that just one extreme value (outlier) can change the value substantially. (Since it cannot resist substantial changes caused by extreme values, we say that it is not a resistant measure of center.)

Midrange

In practice, the __________ is rarely used, but it has three redeeming features: 1. It is very easy to compute. 2. It helps reinforce the very important point that there are several different ways to define the center of a data set. 3. Its value is sometimes used incorrectly for the median, so confusion can be reduced by clearly defining it along with the median.

Average

Never use the term __________ when referring to a measure of center. This word is often used for the mean, but it is sometimes used for other measures of center. To avoid any confusion, we use the correct and specific term, such as mean or median.

∑ (uppercase sigma)

The Greek letter __________ denotes the sum of data values.

Mean from Frequency Distribution

The __________ is equal to the sum of the products of each frequency and midpoint divided by the sum of the frequencies.

Median

The __________ of a data set is the measure of center that is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude.

Mode

The __________ of a data set is the value that occurs with the greatest frequency.

Mean (Arithmetic Mean)

The __________ of a set of data is the measure of center found by adding the values and dividing the total by the number of data values.

x-tilde (x~)

The median is often denoted by __________. To find the median, first sort the values (arrange them in order), and then follow one of these two procedures: 1. If the number of data values is odd, the median is the number located in the exact middle of the sorted list. 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers in the sorted list.

Sample Size

The symbol n denotes the __________, which is the number of data values.

Weighted Mean

When calcul

Round

When calculating measures of center, we often need to __________. We use the following rules: 1. For the mean, median, and midrange, carry one more decimal place than is present in the original set of values. 2. For the mode, leave the value as is without rounding (because values of the mode are the same as some of the original data values).

Bimodal

When two data values occur with the same greatest frequency, each one is a mode and the data set is __________.

μ

__________ = ∑xN is the mean of all values in a population.

x-bar

__________ = ∑xn is equal to a set of sample values.

Mean

__________ is equal to ∑x/n Or, sum of all data values divided by number of all data values.

x

__________ is the variable usually used to represent the individual data values.

N

__________ represents the number of data values in a population.

n (lowercase)

__________ represents the number of data values in a sample.