Measures of Central Tendency

Measures of Central Tendency

Mode

• May have no mode, one mode, or more than one mode. Two modes is called bimodal.

• Mode is not stable. Changing one number can dramatically change the mode.

• Useful when we want to know the most frequently occurring data value such as most requested song, shoe size, most watched show.

• May help to order the data if it is a large group.

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Multiple Choice

What is the mode of this data set?

• 10

• 7

• 18

• 9

Multiple Choice

At Donald's Donuts the number of donut holes in a bag can vary. Help Donald find the MODE.

12, 10, 10, 11, 8, 5, 13, 10, 9

• 10

• 13

• No Mode

Multiple Choice

The mode is the number ________________________.

• you see most

• you see least

• you don't see

• you see first

Fill in the Blank

The number of college credits being taken by a sample of 14 full-time college students for a semester are listed below. The mode is .

12 14 16 15 13 14 15 18 16 16 12 16 15 17

Median

• The median is the central value of an ordered distribution. An equal number of data values above and below the median.

• Uses position rather than the specific value of each data entry.

• If the data has extreme values, the median usually does not change.

• Often used as the average for house prices. One expensive home in the neighborhood wouldn't affect the median selling prices of the other homes.

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​

​Find the median without the 35 cents. One ounce of chips is considered a small serving. Is it reasonable to budget about $10.45 to serve the barbecue-flavored chips to 55 people?

Fill in the Blank

The number of college credits being taken by a sample of 14 full-time college students for a semester are listed below. The median is .

12 14 16 15 13 14 15 18 16 16 12 16 15 17

Fill in the Blank

The Law School Admission Test (LSAT) scores for a sample of seven students accepted into a law school is listed below. The median score is .

174 172 169 176 169 170 175

Multiple Choice

Some quantitative data sets do not have medians.

• True

• False

Multiple Choice

It doesn't matter if the data set is ordered when finding the median.

• True

• False

​For large ordered data sets, the middle of the data set can be found by using the formula below.

​

For an ordered data set of size n,

​

Position of the middle value = ​

Examples:

​

If n = 99​, then the middle value is in the ______position.

​

If n = 100, then the middle value is in the ______ position.​

​Answers:

​

If n =99​ → 50th position

If n = 100 → ​50.5th position (position ≠ value)

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Mean

• Most important of the three. It can be affected by exceptional values.

• μ - pronounced mew (population parameter)

• ∑ - sigma - means sum the following

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Example:

To graduate, Linda needs at least a B in biology. She did not do very well on her first three tests; however, she did well on the last four. Her scores are:

​

58 67 60 84 93 98 100​

​

Compute the mean and determine whether Linda's grade will be a B (80 - 89)​ or C (70 to 79).

Fill in the Blank

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (degrees Fahrenheit) were taken from May to November in the vicinity of Furnace Creek? Find the mode.

Fill in the Blank

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (degrees Fahrenheit) were taken from May to November in the vicinity of Furnace Creek? Find the median.

Fill in the Blank

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (degrees Fahrenheit) were taken from May to November in the vicinity of Furnace Creek? Find the mean.

Resistant Measure

• One that is not influenced by extremely high or low data values.

• Mean is affected by changing the size of only one data value. It IS NOT resistant.

• Median is more resistant. It is not, however, sensitive to the specific size of a data value.

• The trimmed mean is more resistant than the mean, but is still sensitive to the specific data values.

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Trimmed mean

• A trimmed mean is the mean of the data values left after "trimming" a specified percentage of the smallest and largest data values from the data set.

• Usually a 5% trimmed mean is used. (Trim the lowest 5% and the highest 5% of the data. Sometimes a 10% trimmed mean is used.

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​Here 5% of 20 is 1

(5%→.05)

(20 * .05 = 1)​

This means you remove the first number and the last number. (if the numbers were arranged from least to lowest)

Then, you compute the mean without those numbers. This helps to eliminate outliers. ​

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Relationships and Distributions

• ​Mounded - symmetrical or uniform - mean, median, and mode are all the same or close to the same value.

• Left skewed (look at the tail) - mean, median, mode (smallest to largest)

• Right skewed (look at the tail) - mode, median, mean (smallest to largest)​

Poll

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (° F) were taken from May to November in the vicinity of Furnace Creek?

The mean was 167.3° F, the median was 171°F, and the mode was 178°F. What type of distribution exists?

• symmetrical

• Left Skewed

• Right Skewed

Poll

How large is a wolf pack? The following information is from a random sample of winter wolf packs in regions of Alaska, Minnesota, Michigan, Wisconsin, Canada, and Finland (Source: The Wolf, by L.D. Mech, University of Minnesota Press). What type of distribution exists?

• symmetrical

• Left Skewed

• Right Skewed

Weighted Average

• Sometimes we want more importance, or weight, to some of the numbers when averaging.

• How our grades are averaged.

• Multiply each data value by the weight.

• Add the products.

• Divide the by the sum of the weights.​

Weighted Average