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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

Alg1B 9.2 Mean


  • 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”

Alg1B 9.2 Population Venn Mean

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

Alg1B 9.2 Mean Weighted


  • 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.

Alg1B 9.2 Median


  • 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
    • Answer: 3

Alg1B 9.2 Outliers

 

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.

Alg1B 9.2 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

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