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Chapter 3Descriptive Measures

Section 1

Measures of Center

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Descriptive Measures

Numbers that are used to describe data sets are called descriptive measures.

Statistical methods can be used to summarize data

Measures of average are also called measures of central tendency or measures of center and include the mean, median, mode, and midrange. Indicate where the center or most typical value of the data set lies.

Measures that determine the spread of data values are called measures of variation and include the range, variance, and standard deviation.

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General Rounding Rule

In statistics the basic

rounding rule is that when

computations are done in the

calculation, rounding should

not be done until the final

answer is calculated.

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Mean of a Data Set

The mean of a data set is the sum of the observations divided by the number of observations.

Mean is known as the arithmetic average.Found by adding the values (Sum of values) of the data and dividing by the total number of values.

The symbols used are the Greek letter μ (mu) for the population and (x-bar) for the sample.X

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The Mean:

• Compute the mean by using all the values of the data.

• Varies less than the median or mode when samples are taken from the same population and all three measures are computed for these samples.

• Is used in computing other statistics, such as variance.

• The data set is unique, and not necessarily one of the data values.

• Cannot be computed for an open-ended frequency distribution.

• Is affected by extremely high or low values and may not be the appropriate average to use in these situations.

n

x

n

xxx n

...1

N

X

N

XXXX N

...321Population Sample

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Table 3.1 Table 3.2Data Set I Data Set II

Example 3.1Professor Hassett spent one summer working for a small mathematical consulting firm. The firm employed a few senior consultants, who made between $800 and $1050 per week; a few junior consultants, who made between $400 and $450 per week; and several clerical workers, who made $300 per week. The firm required more employees during the first half of the summer than the second half. The tables list typical weekly earnings for the two halves of the summer.

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Table 3.4

Solution Example 3.1

Interpretation: The employees who worked in the first half of the summer earned more, on average (a mean salary of $483.85), than those who worked in the second half (a mean salary of $474.00).

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Median of a Data Set

Median – is the middle point in the data set that has been ordered. The symbol for the median is MD.

Arrange the data in increasing order.

• If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list.

• If the number of observations is even, then the median is the mean of the two middle observations in the ordered list.

In both cases, if we let n denote the number of observations, then the median is at position (n + 1) / 2 in the ordered list.

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The median: is the halfway point (middle value) in a data set when the data is in

ascending/descending order.

is also a measure of the center.

is found by arranging the data in order and selecting the middle point.

The symbol for the median is MD.

is used to find the center or middle value of a data set.

is used when we must determine whether the data values fall into the upper half or lower half of the distribution.

is used to find the average of an open-ended distribution.

is affected less than the mean by extremely high or extremely low values.

either a specific value or may fall between two values.

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Steps in finding the median

1. Arrange the data in order

2. Select the middle point

OR

If even number of data values, add the two middle values and divide by 2.

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Mode of a Data Set

Mode – is the most frequent data value. There is no symbol used.

Find the frequency of each value in the data set.

• If no value occurs more than once, then the data set has no mode.

• Otherwise, any value that occurs with the greatest frequency is a mode of the data set.

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The mode:

• The value that occurs most often in a data set.

• is used when the most typical case is desired.

• is the easiest average to compute.

• can be used when the data are nominal, such as religious preference, gender, or political affiliation.

• is not always unique. A data set can have more than one mode, or the mode may not exist for a data set.

• The mode for grouped data is the modal class and is the class with the largest frequency.

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Notations & Formulas

X

n

x

n

xxx n

...1

x variable for which we have sample data

n number of values in the sample sizex1 first observation ...xn nth observation∑ Sum of values. Greek letter sigma

sample mean. x-barμ population mean. Greek letter muN number of values in the population

Sample Mean

N

X

N

XXXX N

...321Population Mean

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The mean of the population is the average of all values of x in the population and is denoted by .

Greek letter mu

may vary from sample to sample. has only one value.

Always assume that data are obtained from a sample unless otherwise specified.

x

The type of mean that considers an additional factor is called the weighted mean. It is used when the values are not equally represented.

Rounding rule for the Mean: the mean should be rounded to one more decimal place than occurs in the raw data.

x-bar

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Figure 3.1

This figure shows the relative positions of the mean and median for right-skewed, symmetric, and left-skewed distributions. Note that the mean is pulled in the direction of skewness, that is, in the direction of the extreme observations. For a right-skewed distribution, the mean is greater than the median; for a symmetric distribution, the mean and the median are equal; and, for a left-skewed distribution, the mean is less than the median.

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A resistant measure is not sensitive to the influence of a few extreme observations

The median is a resistant measure of center, but the mean is not.

A Trimmed mean can improve the resistance of the mean: to do this remove the percentage of the smallest and largest observations before computing the mean.

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Sample Mean

Sample Mean is the mean of the sample data.

There are many sample means – one for each possible sample of the population.

n

x

n

xxx n

...1

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Population Mean

N

X

N

XXXX N

...321

Population Mean or Mean of the variable is the mean of the population data.

There is only one population mean – namely, the mean of all the possible observations of the variable for the entire population.

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Example:Burglaries at Pennsylvania University. The following data are the number of burglaries reported for a specific year for nine western Pennsylvania universities. Find the mean, median, and mode. Which measure of average might be the best in this case? Explain. 61, 11, 1, 3, 2, 30, 18, 3, 7

1.159

136X

n

XMean =

Median = 1, 2, 3, 3, 7, 11, 18, 30, 61

Mode = most repeated = 3

I think the mean is the best measure. The median and mode are to small a number for the sample and who give the impression of less burglaries.