Measures of Central Tendency

Measures of Central Tendency

By Anita Langston

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

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?

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

Multiple Choice

If a data set has an even number of data, is it true or false that the median is never equal to a value in the data set? Explain.

• True

• False

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

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For an ordered data set of size n,

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Position of the middle value = ​

Examples:

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

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

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:

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58 67 60 84 93 98 100​

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

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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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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

Example:

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Suppose your midterm test score is 83 and your final exam score is 95. Using weights of 40% for the midterm and 60% for the final exam, compute the weighted average of your scores.

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Do you have an A?​

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