Statistics Chapter 5

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damea134
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280236
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Statistics Chapter 5
Updated:
2015-08-09 23:29:35
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Data Approximation
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Normal Approximation for Data
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  1. Areas Under the Approximation Cruve
    • -1 and +1 is about 68% out of all entries 
    • -2 and +2 is about 95% out of all entries
    • -3 and +3 is about 99.7% out of all entries
    •  
    • anything under or over -3 and +3 is about 6/100,00. 

    68% is the area under the Curve from -1 to +1 

    95% is the area under the curve from -2 to +2 

    Note: there is no (0) values just keep getting smaller and smaller
  2. Stander approximation Cruve
    1. It is Symmetric at 0

    2. the Total area under the curve = 100%

    3.  It's always about the horizontal axis
  3. Stander Units
    A value is converted to standard units by seeing how many SDs it is above or below the average

    Standard units range from 0 (0% area) to 4.45 (99.991 area).

    Standard units are used when using the Standard Approximation Curve.
  4. Converting Values to Standard Units
    To Convert a value to a Standard Unit you will need to know the SD and the Average

    If the SD is 3 and the Average is 70 then 73 or 67 would be +-1 Standard Unit away. 

    Example 70 + 3 (1SU) = 73 

    70 - 3 (1SU) = 67

    to Convert an SD into the SU than you need to Divide the SU by the SD

    If the number is 72 then 72-70 = 2, 2/3 =(.67 SU) 

    If number is 78 than 78-70 = 8, 8/3 = (2.67 SU)

    to Convert an SU into the SD than you need to multiple the SD by the SU

    Example: 2.66 Standard Unit, SD = 3

    (2.67 SU) X (3 SD) = 8
  5. Finding numeral value of an Standard Units
    If Standard Unit is 2.5, -2.7, -.67 and the average is 50 and the SD is 10 then you have to multiply the SD by the SU and add the sum by the Average (50)

    2.5(SU) x 10(SD) = 25, 25 + 50(Ave)= 75

    -2.7(SU) x 10(SD) = -27, 50(Ave) -27= 33

    -.67(SU) x 10(SD) = -6.7, 50(Ave) -6.7= 53.3
  6. Finding Areas of Standard Units
    The Normal Standard Unit Table/Chart gives the area of Standard Units

    • 0 SU= 0% Area
    • 1 SU= 68% Area 
    • 2 SU= 95% Area
    • 3 Su= 99.73% Area
    • 4 SU= 99.99% Area

    Note Areas are given from +1to-1(68%). The full area under the curve is 100%
  7. Calculating Areas Under Standard Unit Curve.
    To find the area under the SU curve you must know the SU Area Percentage Value which is given on Standard Unit Table/Chart

    Find the Area to the right of 1 SU

    1 Su = 68% area, 68-100% = 32%,

    (Note: 32% is for areas to the right and left of 1. we just want the area to the right)

    to get area to the right divide 32% by 2

    32%/2= 16% (answer) 

    If it is said find area to the left of 1SU than you would add 16% to 68% to get total area.  (16%+68%) = 84% would be the area to the left of 1. 

    Be sure to draw diagrams of the curve and isolate the area in which you are looking for the percentage.
  8.  to Convert an SD into the SU

    Important
    to Convert an SD into the SU than you need to Divide the SU by the SD

    If the number is 72 then 72-70 = 2, 2/3 =(.67 SU) If number is 78 than 78-70 = 8, 8/3 = (2.67 SU)
  9. Normal Approximation for data

    Calculating areas using real data
    Average height of men is 69 inches and have a SD of 3. Estimate percentage of heights between 63 and 72

    Step 1. Convert values SU. 

    • 66-69= -6/3 SD = -2 SU, +-2 SU = 95% Area
    • 72-69= 3/3 SD = 1 SU, +-1 SU = 68% Area

    2. Sketch line of desired area

             -2_________0__________1

    3. Calculate for area

    • -2 =95%/2 =47.5%
    • 1 = 68%/2 = 34% 

    Area= 81.5%
  10. Calculating Areas
    There are two ways to calculate areas

    If the area is in between two points (-1, and 2) than you would take both areas and divide them by two (68%/2 and 95%/2) and add the sums (34%/2 - 47.5%/2 = 81.5).  

    When points are between two areas it usually involve adding and subtracting the half of the area (1=68/2=34%) and (2=95/2=47.5%).

    If a point deals with areas to the left, right, or on the ends than adding up areas and subtracting by 100% Will get the answer.
  11. Interquartile Range

    (Percentiles that Don't follow Normal Curve)
    The difference between the 75th and 25th percentile is called the interquartile range. It is a useful way to quantify scatter.

    Interquartile Range= 75th percentile - 25th percentile.

    this is sometimes used as a measure of spread when the distribution has a long tail.
  12. Adding Percentiles

    (Percentiles that Don't follow Normal Curve)
    • %    Range 
    • 1     $0
    • 9     $1,000
    • 10   $5,000
    • 25   $10,000
    • 25   $12,000
    • 50   $24,000
    • 75   $100,000
    • 90   $110,000
    • 99   $120,000

    If I wanted to find out the 75th distribution of income, I would add up all of the Ranges to the 75th percentile range.

    • Example:
    • 0+$1,000+5,000+$10,000+$12,000+ $24,000+$10,0000 = $62,000

    The 75th percentile would = $62k
  13. Calculating Area of Percentiles

    (Percentiles That follow Normal Curve)
    Calculating a math score

    Math Scores average = 535, SD=100

    Estimate the 95th Percentile Range.

    95% of a Curve in SU is going to be 90%

    • Calculation:
    • the 95% is between +-90% (or +-1.65) We want area all the way to the right of 95%.
    • To get the other area to the right you would have to subtract 5% (100%-95% = 5%). leaving you with 90%

    90% area = 1.65 SD

    To get the score you would multiply 1.65 SU by the SD of 100, and add it to the average (535)

    1.65x 100 = 165 +535(ave) = 700


    • Note:
    • A percentile is a score 95th percentile = 700 

    A percentile rank, however is a percentage. If you score 700 on a test it puts you in the 95% percentile rank

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