# 9.2 – Measures of Center

## Key Terms

• Mean – The sum of a set of values divided by the number of values.
• The mean is also called the average.
• Measures of Central Tendency – Measures of the center or middle of a data set.
• Median – The middle value in a data set. Half the values are below the median and half are above the median.
• Mode – The data value that appears most often in a data set.
• When more than one data value appears equally often and more often than any other values, each is a mode.
• Outlier – A data value that is markedly different from most of the other data values in a sample.
• Population – The entire group being considered.
• Sample – A selection of items chosen from a larger population.

## Notes

Mean (Average)
• You can find the mean by adding up all of the values in a data set and dividing that sum by the number of values that were added.
• $mean\;=\;\frac{x_1\;+\;x_2\;+\;...\;+\;x_n}{n}$
• Ex. For 4, 5, 34, 34, 44, 53
• $mean\;=\;\frac{4\;+\;5\;+\;34\;+\;34\;+\;44\;+\;53}{6}$
• $mean\;=\;\frac{174}{6}$
• $mean\;=\; 29$
• Mean of a Sample (of Population)
• Symbol $\overline{x}$
• Pronounced: x-bar

• Ex For the mean of 4, 5, 34, 34, 44, 53
• Write the formula: $\overline{x}=\frac{1}{n}\sum\limits_{i=1}^n x_i$
• Substitute 6 for n since there are 6 values in the data set: $\overline{x}=\frac{1}{6}\sum\limits_{i=1}^6 x_i$
• Substitute the sum of the data: $\overline{x}=\frac{1}{6}\;\;(4\;+5\;+34\;+34\;+44\;+55)$
• Simplify: $\overline{x}\;=\;\frac{174}{6}$
• Answer: $\overline{x}\;=\; 29$

• Mean of Population
• Symbol: µ
• Pronounced “myew” or “mew”

Weighted Mean
• When you take the mean of a set of numbers, each value has the same weight. But sometimes, you want certain values to be weighted.
• Ex. Tests and quizzes are worth points, but tests may be weighted more heavily (making them affect the final grade more).
• Tests may be weighted at 60% and quizzes may be weighted at 40% even though point values may be 100 points for each.
• Formula for Finding the Weighted Mean

• Ex. Let’s say you are playing a video game where you must collect items. Your score will be averaged, depending on the item’s weight.
• Hearts are worth 5 points.
• Fruit is worth 3 points.
• Rocks are worth 1 point.
• If you get 6 hearts, 2 pieces of fruit, and 18 rocks, what’s your score?
• $weight\; \bullet \; number$, then divide by total weights.
• The total weights are 5 + 3 + 1 = 9.
• Setup: $\frac{5\;\bullet \;6\;+\;3\;\bullet \; 2\;+\;1\; \bullet \; 18}{9}$
• Simplify: $\frac{30\;+\;6\;+\; 18}{9}$
• Simplify: $\frac{54}{9}$
• Answer: You scored a 6 in this round!

Median
• To find the median
1. Arrange all numbers from smallest to largest.
2. If n is odd, the median is the middle number.
3. If n is even, the median is the average (mean) of the two middle terms.

• Ex. A baseball stadium offers tickets with different prices, depending on the location of the seats. If it increases the prices of all tickets by $30, what will happen to the median price of the tickets? • Answer: It will increase by$30

Mean & Median with an Outlier
• An outlier is a data value that is pretty far from the rest.
• In general, the median of a data set is more resistant to outliers than the mean.
• Ex. What is the outlier in this set: 45, 3, 53, 29, 63, 33

Mode
• The mode is the data value that is repeated most often.
• If no data value is repeated more than once, there is no mode.
• If multiple data values occur an equally high number of times, they are each a mode.

## Examples

 Ex 1. What is the mean of the following data values?  22, 37, 49, 15, 92 Setup: $\frac{22\;+\; 37\;+\; 49\;+\; 15\;+\; 92}{5}$ Simplify: $\frac{215}{5}$ Answer: 43 Ex 2. If the formula $\overline{x}=\frac{1}{n}\sum\limits_{i=1}^n x_i$ is used to find the mean of the following sample, what is the value of n? 2, 63, 88, 10, 72, 99, 38, 19 There are 8 items in the data set, so n = 8 Ex 3. If a sample of 187 bowlers were taken from a population of 999 bowlers, $\overline{x}$ could refer to the mean of how many bowlers’ scores? Answer: 187 Ex 4. If a sample of 21 runners were taken from a population of 580 runners, µ could refer to the mean of how many runners’ times? Answer: 580. Ex 5. If there are 12 values in a data set, in order from smallest to largest, what is the median of the data set? Answer: the mean of the 6th and 7th value. Ex 6. What is the mode of the following data values?  87, 19, 11, 52, 11 Answer: 11, because it appears the most number of times of any of the values in the data set. Ex 7. An ice cream shop’s goal is to sell a mean of 45 ice cream cones per day. In one week, it sold the following numbers of ice cream cones each day. 54, 45, 33, 39, 48, 40, 21 How many more ice cream cones would it have had to sell during the week to meet its goal? Setup: goal = sales plus the unknown, divided by the 7 days that are in the week $45=\frac{54\;+\; 45\;+\; 33\;+\; 39\;+\; 48\;+\; 40\;+\; 21\;+\; x}{7}$ Multiply by 7 on both sides and simplify: $315=280\;+\; x$ Answer: 35 Ex 8. Cora is taking three AP classes and two regular classes. Her AP classes count twice as much as her regular classes in her GPA. Each A is worth 4 points, Bs are worth 3 points, Cs are worth 2 points, and Ds are worth 1 point. What is Cora’s GPA? Classes & Grades AP English: A AP Government: A AP Algebra II: A Spanish: B Physics: A Note: Since AP classes count as 2, the total weights are 8 Setup: $\frac{2\; \bullet \;4 \; +\; 2\; \bullet \;4 \; +\; 2\; \bullet \;4 \; +\; 1\; \bullet \;3 +\; \;1\; \bullet \;4}{8}$ Simplified: $\frac{8\; +\; 8\; +\; 8\; +\; 3\; +\; 4}{8}$ Answer: 3.875, rounded to 3.9 GPA