Unifix cubes activity Get into groups of 6-10 people. Each person in the group grab a handful of unifix cubes. It can be a large handful or a small

MEASURES OF CENTRAL TENDENCY

Unifix cubes activity Get into groups of 6-10 people. Each person in the group grab a

handful of unifix cubes. It can be a large handful or a small handful. A variety is good.

Without paper, pencil, or calculator: devise a plan for redistributing the cubes in your group so that everyone has the same number of cubes.

Devise a second, possibly more efficient plan.

Mean Sample Mean

The sample mean is the arithmetic mean of a set of sample data, given by

or

where xi is the ith data value and n is the number of data values in the sample.

Mean as a balancing point

Get into groups of 4-6. Find the balancing point of the following

numbers on a number line using post-it notes:

3, 4, 4, 4, 4, 5, 5, 6, 7, 8, 8, 10, 10 Try it again with:

2, 3, 3, 3, 4, 4, 4, 4, 5, 8 , 10, 10 Find a list of 10 numbers that have a

mean of 7 using mean as a balancing point.

Median Finding the Median of a Data Set

1. List the data in ascending (or descending) order, making an ordered array.2. If the data set contains an ODD number of values, the median is the middle value in the ordered array. 3. If the data set contains an EVEN number of values, the median is the arithmetic mean of the two middle values in the ordered array. Note that this implies that the median may not be a value in the data set.

Calculating Measures of Center—Mean, Median, and Mode

Given the recent economy and change of attitude in society, many people chose to take on another job after retiring from one. Below is a sample of ages at which people truly retired; that is, they stopped working for pay. Calculate the mean, median, and mode for the data.

84, 80, 82, 77, 78, 80, 79, 42

Calculating Measures of Center—Mean, Median, and Mode (cont.)

Mean: Remember, the mean is the sum of all the data points divided by the number of points.

84 80 82 77 78 80 79 428

60275 75..25 3

8

ixxn


Median: We have an even number of values, so we will need the mean of the middle two values in the ordered array.

42, 77, 78, 79, 80, 80, 82, 84 79 80

279.5


Mode: The number 80 occurs more than any other number, so it is the mode.

Choosing an Appropriate Measure of Center

Determining the Most Appropriate Measure of Center

1.For qualitative data, the mode should be used.

2.For quantitative data, the mean should be used, unless the data set contains outliers or is skewed.

3.For quantitative data sets that are skewed or contain outliers, the median should be used.

Determining Mean, Median, and Mode from a Graph

Determine which letter represents the mean, the median, and the mode in the graph below.

Graphs and Measures of Center

Graphs and Measures of Center 1. The mode is the data value at which a distribution has its highest peak.2. The median is the number that divides the area of the distribution in half. 3. The mean of a distribution will be pulled toward any outliers.

MEASURES OF DISPERSION

Range

Range Range = Maximum Data Value −

Minimum Data Value

Standard Deviation Standard Deviation

The standard deviation is a measure of how much we might expect a typical member of the data set to differ from the mean.

The population standard deviation is given by

Standard Deviation Standard Deviation (cont.)

where xi is the ith value in the population,

μ is the population mean, and

N is the number of values in the population.

Standard Deviation Standard Deviation (cont.)

The sample standard deviation is given by

where xi is the ith data value, x̄ is the sample mean, and n is the number of data values in the sample.

Calculating Standard Deviation

Calculate the sample standard deviation of the following data collected regarding the numbers of hours students studied for a physics exam.

5, 8, 7, 6, 9 Solution Let’s calculate the sample standard deviation by hand using the following formula.

Calculating Standard Deviation (cont.)Deviations and Squared Deviations of the Data

xi (xi – x̄) (xi – x̄)2

5 5 − 7 = −2 48 − 7 = 1 17 7 − 7 = 0 06 6 − 7 = −1 19 9 − 7 = 2 4

Next, find the sum of the squared deviations by adding up the values in the last column.

Calculating Standard Deviation (cont.)

Finally, substitute the appropriate values into the sample standard deviation formula as follows.

Interpreting Standard Deviations

Mark is looking into investing a portion of his recent bonus into the

stock market. While researching different companies, he discovers the

following standard deviations of one year of daily stock closing prices.

Profacto Corporation: Standard deviation of stock prices =

$1.02

Yardsmoth Company: Standard deviation of stock prices =

$9.67

What do these two standard deviations tell you about the stock prices

of these companies?

Interpreting Standard Deviations (cont.)

Solution

A smaller standard deviation indicates that the data values are closer together, while a larger standard deviation indicates that the data values are more spread out. In this example, the standard deviation of stock prices for the Profacto Corporation is considerably smaller than that of the Yardsmoth Company. Hence, there is less variability in the daily closing prices of the Profacto stock than in the Yardsmoth stock prices.

Interpreting Standard Deviations (cont.) If Mark wants a stable long-term investment, then Profacto appears to be the better choice. If, however, Mark is looking to make a quick profit and is willing to take the risk, then the Yardsmoth stock would seem to better suit his purposes. Note that looking at the standard deviations is just one component of evaluating market prices.

Empirical Rule Empirical Rule for Bell-Shaped

Distributions •Approximately 68% of the data values lie within one standard deviation of the mean.

•Approximately 95% of the data values lie within two standard deviations of the mean.

•Approximately 99.7% of the data values lie within three standard deviations of the mean.

Applying the Empirical Rule for Bell-Shaped Distributions

The distribution of weights of newborn babies is bell-shaped with a mean of 3000 grams and standard deviation of 500 grams.

a. What percentage of newborn babies weigh between 2000 and 4000 grams?

b. What percentage of newborn babies weigh less than 3500 grams?

c. Calculate the range of birth weights that would contain the middle 68% of newborn babies’ weights.

Applying the Empirical Rule for Bell-Shaped Distributions (cont.)

Thus, these weights lie two standard deviations above and below the mean. According to the Empirical Rule, approximately 95% of values lie within two standard deviations of the mean. Therefore, we can say that approximately 95% of newborn babies weigh between 2000 and 4000 grams.


b. To begin, let’s find out how many standard deviations a weight of 3500 grams is away from the mean by performing the same calculation as before.


Thus it is one standard deviation above the mean. The Empirical Rule says that 68% of data values lie within one standard deviation of the mean. Because of the symmetry of the distribution, half of this 68% is above the mean and half is below. Putting the upper 34% together with the 50% of data that is below the mean, we have that approximately

of newborn babies weigh less than 3500 grams.


Applying the Empirical Rule for Bell-Shaped Distributions (cont.) c. From the Empirical Rule, we know

that 68% of the data values lie within one standard deviation of the mean for bell-shaped distributions. The standard deviation of this distribution is 500; thus, by adding 500 to and subtracting 500 from the mean of the distribution, we will get the range of birth weights that contain the middle 68% of newborn babies’ weights.


Upper end: 3000 + 500 = 3500 Lower end: 3000 − 500 = 2500

Thus, 68% of newborn babies weigh between 2500 and 3500 grams.

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Unifix cubes activity Get into groups of 6-10 people. Each person in the group grab a handful of unifix cubes. It can be a large handful or a small