1.3 Measures of Center


Simplify. Round to the tenths place.


  1. \(\frac{32+97}2\)

    \(129/2\)

    \(64.5\)

  2. \(\frac{120+162+95+144+135+160+154}7\)

    \(970/7\)

    \(138.6\)

  3. \(\frac{3*55 + 9*65 + 4*75}{16}\)

    \(1050/16\)

    \(65.6\)

  4. \(\frac{75.8 - 53.9}2\)

    \(21.9/2\)

    \(11\)

    5. Mean, Median, Mode, Midrange


    What is an arithmetic mean? The average value for a set of numbers


    What is the formula for the mean of a sample? \(\overline{x}=\frac{\sum x}{n}\)


    Describe in words what the formula for sample mean tells you to do: Take the sum of the numbers in the data set and divide by the number of data values.


    What do each of the symbols in the formula represent? \(\overline{x}\) is the sample mean; \(\sum\) denotes the sum of a set of values; \(x\) is an individual data value; \(n\) represents the number of data values in the sample


  5. The mean weight of three dogs is 60 pounds. Which of the following is possible? (Answer yes or no.)
    • One of the dogs weighs 100 pounds. yes
    • None of the dogs weighs more than 60 pounds. yes
    • Each dog weighs less than 60 pounds. no
    • The dogs weigh 25, 55, and 110 pounds. no
    • The dogs weigh 24, 52, and 104 pounds. yes
    • Altogether the dogs weigh 180 pounds. yes
  6. Shanita made a note of the weights of 8 baskets of fruit, but she spilled coffee on the fruit and the last number got smudged. If the average weight of the baskets is 12.5 pounds, how much was the smudged number? In other words, how much did the 8th basket of fruit weigh?
    A chart showing the weights of all of the baskets of fruit. Basket 1 is 10.8 pounds, Basket 2 is 9.3 pounds, Basket 3 is 14.8 pounds, Basket 4 is 12.9 pounds, Basket 5 is 13.1 pounds, Basket 6 is 10.6 pounds and Basket 7 is 15.3 pounds. Basket 8 is covered in a coffee spill and we cannot read the weight of the basket.

    Sum of basket weights is \(8*12.5 = 100\), so basket 8 must be \(100 – 86.8 = 13.2\)

    7. Define the following measures of central tendency:

    Median: the middle value in an ordered data set

    Mode: the value that occurs the most often in a data set

    Midrange the mean of the max and min values in the data set

     

  7. Calculate the measures of central tendency for the following data.
    1. \(0, 3, 5, 6, 6, 8, 9, 10, 10, 10\)

      mean: \(67/10 = 6.7\)

      median: \(7\)

      mode: \(10\)

      midrange: \(0+10/2=5\)

    2. \(0, 3, 5, 6, 6, 8, 9, 10, 10, 100\)

      mean: \(157/10 = 15.7\)

      median: \(7\)

      mode: \(6, 10\) (bimodal)

      midrange: \(0+100/2=50\)

  8. Calculate the measures of central tendency for the life expectancies from the given data.

    Country Health $ per capita Obesity % Life expectancy Universal Healthcare
    Canada 5292 30.1 82 yes
    China 420 7.3 76 no
    Germany 5411 22.7 81 yes
    Italy 3258 23.7 84 yes
    Japan 3703 3.5 84 yes
    Mexico 677 27.6 77 no
    Norway 9522 24.8 82 yes
    Switzerland 9674 21 83 yes
    United Kingdom 3377 29.8 82 yes
    United States 9403 35 79 no

    1. Mean: 81
    2. Median: 82
    3. Position of the Median: \(\frac{n+1}{2}=\frac{10+1}{2}=5.5\)
    4. Mode: 82
    5. Midrange: \(\frac{76+84}{2}=80\)
  9. Using the table in the problem above, calculate the measures of central tendency for the obesity percentages from the given data. Verify using technology.
    1. Mean: 22.55
    2. Median: 24.25
    3. Position of the Median: \(\frac{n+1}{2}=\frac{10+1}{2}=5.5\)
    4. Mode: none
    5. Midrange: \(\frac{3.5+35}{2}=19.25\)


    10. Shapes of Distributions

    Symmetric (Normal):

     Mean and Median are the same as Mode

    A picture of a histogram and a picture of a curve drawn around the histogram with the bars removed. The histogram is labeled 'Histogram (n=21)'.The horizontal axis represents the x values and goes from 0 to 6, counting by 1. The vertical axis represents the frequency and goes from 0 to 10, counting by 2. The bar at 1 has a frequency of 2. The bar at 2 has a frequency of 5. The bar at 3 has a frequency of 7. The bar at 4 has a frequency of 5. The bar at 5 has a frequency of 2. The horizontal axis for the curve represents the x values and goes from 1.5 to 7.5, counting by 1. There is no vertical axis and the curve is a normal bell shaped curve.



    Skewed To The Left (negatively skewed):

     Mean and Median are to the LEFT of the Mode (Left Tail)

    A histogram labeled 'Histogram (n=29)'. The horizontal axis represents the x values and goes from 0 to 8, counting by 2. The vertical axis represents the frequency and goes from 0 to 12, counting by 2. The bar at 1 has a frequency of 1. The bar at 2 has a frequency of 3. The bar at 3 has a frequency of 5. The bar at 4 has a frequency of 7. The bar at 5 has a frequency of 9. The bar at 6 has a frequency of 4.

    Skewed To The Right (positively skewed):

     Mean and Median are to the RIGHT of the Mode (Right Tail)

    A histogram labeled 'Histogram (n=28)'. The horizontal axis represents the x values and goes from 0 to 8, counting by 2. The vertical axis represents the frequency and goes from 0 to 12, counting by 2. The bar at 1 has a frequency of 3. The bar at 2 has a frequency of 9. The bar at 3 has a frequency of 7. The bar at 4 has a frequency of 5. The bar at 5 has a frequency of 3. The bar at 6 has a frequency of 1.




    Bimodal:

     Two Modes

    A histogram labeled 'Histogram (n=35)'. The horizontal axis represents the x values and goes from 0 to 6, counting by 1. The vertical axis represents the frequency and goes from 0 to 12, counting by 2. The bar at 1 has a frequency of 5. The bar at 2 has a frequency of 10. The bar at 3 has a frequency of 5. The bar at 4 has a frequency of 10. The bar at 5 has a frequency of 5.

    Uniform:

     All frequencies are the same

    A histogram labeled 'Histogram (n=25)'. The horizontal axis represents the x values and goes from 0 to 6, counting by 1. The vertical axis represents the frequency and goes from 0 to 6, counting by 2. The bar at 1 has a frequency of 5. The bar at 2 has a frequency of 5. The bar at 3 has a frequency of 5. The bar at 4 has a frequency of 5. The bar at 5 has a frequency of 5.

    Weighted Means and Means of Grouped Data

  10. Calculate the semester grade point average (GPA) for the student with the following grades:
    Course Credit Hours Grade Point Value for Grade Quality Points (Points x hours)
    ENGL 1020 3 C

    2

    6
    MATH 1530 3 A

    4

    12
    COMM 1010 3 B

    3

    9
    BIOL 1110 4 C

    2

    8
    COLL 1000 1 B+

    3.5

    3.5
    Total

    14

    38.5

    \(\frac{38.5}{14}=2.75\)



  11. Find the estimated mean of the following data:

    SECONDS FREQUENCY MIDPOINTS
    5.0 – 5.9 5

    5.45
    6.0 – 6.9 8

    6.45
    7.0 – 7.9 10

    7.45
    8.0 – 8.9 7

    8.45
    9.0 – 9.9 3

    9.45

    1. What is the estimated mean of the data? 7.30

      By Formula: \(\overline{x}=\frac{\sum(f \cdot x)}{\sum f}=\frac{5(5.45)+8(6.45)+10(7.45)+7(8.45)+3(9.45)}{33}=\frac{240.85}{33} \approx 7.30\)

    2. What class holds the median number of seconds? 7.0-7.9
    3. What is the shape of the distribution? Normal

  12. Below is a frequency distribution of the ages of riders for one ride on the Dahlonega Mine Train at Six Flags. Find the estimated mean of the following data:

    Age of Riders, in years Number of Riders
    10 - 14 5
    15 - 19 6
    20 - 24 9
    25 - 29 15
    30 - 34 3

    1. What is the estimated mean of the data? 22.7
    2. What class holds the median number of seconds? Class 20-24

      Position of the median: \(\frac{38+1}{2}=\frac{39}{2}=19.5\). Add frequencies until you reach 19.

    3. What is the shape of the distribution? Skewed to the left
  13. The stem-and-leaf plot displays the average life-span of a sample of animals. Key: 1|5 = 15 years
    Stem Leaves
    0 3 4 6 8
    1 0 0 2 2 2 5 5 6 8
    2 0 0 0 0
    3
    4 0
    1. Determine the median life-span of the sample data set. \(\frac{12+15}{2}=\) 13.5 years
    2. What is the mode of the sample data set? 20 years
    3. Calculate the mean of the sample data set. 14.5 years

    A chart with the average life-span of a sample of animals. Baboon 20 years, black bear 18, polar bear 20, camel 12, cat 12, chimpanzee 20, chipmunk 6, cow 15, deer 8, dog 12, elephant 40, guinea pig 4, horse 20, mouse 3, squirrel 10, tiger 16 and zebra 15.

    14. Give One/Get One


    Make a list of everything you know about the measures of central tendency under My List. Think about meanings, formulas, special cases, advantages, disadvantages, etc. When prompted, discuss the terms with a classmate to add to your information and share your information with a classmate.


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