?Statistics and Data Analysis

"Numbers don't lie, but you can lie with numbers"
Numbers may be manipulated to persuade or convince.

SEE: DATA ANALYSIS and CHARTS

Vocabulary:
Data is raw values.
Information is doing something to the data to make meaning of the numbers.

Mean:  Arithmetic Average: SUM of all Numbers / Number of Values. The Mean or average is always bigger than the Minimum, and  Smaller then the Maximum.

Median: Middle Value.  Arrange data values in Order. What is the 'Middle' Value?  If you have an EVEN number of values, you 'average' the two in the middle.

Mode: Most Often Value 
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Low: Minimum:  Smallest Value
High: Maximum: Largest Value
Range: Spread: (High - Low)

Example 1:  10, 12, 20, 24
The Low (Minimum) is: 10
The High (Maximum) is: 24
The Range is: 14 (24 - 10)
Mean (Average) = (10 + 12 + 20 + 24) / 4 = 66/4 = 16.5  Be carful that the result is "Meaningful". If this was "Boys in a class", the mathematical answer is 16.5. However, you cannot have 1/2 a boy!
Median is the Middle. There are 4 data values:  the middle is between 12 and 20: (12 + 20)/2 = 32/2 = 16
Mode: None

Example 2:  75, 77, 80, 84, 94
The Low (Minimum) is: 75
The High (Maximum) is: 94
The Range is: 19 (94 - 75)
Mean (Average) = (75 + 77 + 80 + 84 + 90) / 5 = 410/5 = 82
Median is the Middle. There are 5 data values:  The middle Value Point is (1 + 5)/2 = 6/2 = 3. The Third value in the list is 80.
Mode: None

PAR METHOD for Averaging
Work Smart, not hard!
1) GUESS a value for the Average (maybe use 80 in this example)
2) Find the difference between the Guess Value (80) and each Data Value:
Decide if it is easier to work with Positive or Negative numbers.
       80 - 75 = 5     80 - 77 = 3       80 - 80 = 0      80 - 84 = -4     80 - 94 = -14
or: 75 - 80 = -5     77 - 80 = -3     80 - 80 = 0      84 - 80 = 4       84 - 80 = 14
3) Find the SUM of the Differences: (5 + 3 + 0 - 4 - 14) = 10
4) Find the 'average' difference: 10 / 5 = 2. 
5) Add this result to the Target (Guessed Average: 80):  80 + 2 = 82

A problem that is often asked requires you to find a value that gives an AVERAGE:
For Example: Mary scored: 84, 95, and 80 on three tests.  What does she need to score on the forth test to have an average greater than 88?
Since the Average = (SUM / #)  or  88 = SUM / 4.   The sum of the scores is 88 x 4 = 352.
84 + 95 + 80 = 259.
352 - 259 = 93.  

USING THE PAR Method.  The Desired AVERAGE is 88.
88-84 = -4       95-88 = 7       80-8 = -8.   So: -4 + 7 - 8  = -4 - 8 + 7 = -5.
She is 5 points away from her goal so she needs to 'Gain +5' on the last test.  88 + 5 = 93.




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FREQUENCY TABLES and DATA
Be careful when data is provided in a FREQUECY TABLE:
The data may be in order by Frequency, it may be in order by Value 
Frequency      Value
      4                    6
      2                    3
      2                    5
      1                     8

If it is not given in the problem, you need to add up all the numbers in the Frequency Column. There are nine (Nine) data values for this problem.

The MODE is the number that appears MOST OFTEN:  there are 4 Sixes.

To find the AVERAGE or MEAN it is not: (6 + 3 + 5 + 8) / 4  and it is not: (6 + 3 + 5 + 8) / 9

It is the SUM / #:  (6 + 6 + 6 + 6) + (3 + 3) + ( 5 + 5) + 8 or (4x6 + 2x3 + 2x5 + 1x8) = 48 /9

For the MEDIAN, The values need to be in order:  Since there are 9 data values, the middle value is (1 +9)/2 or the 5th position will be the MEDIAN.
If there were an EVEN number of data points: Say you had 12 Values: (1 + 12) / 2 = 6.5.
You would need to AVERAGE the data values at Position 6 and Position 7 to find the Median Value.
Putting the Data in Order: 3, 3, 5, 5, 6, 6, 6, 6, 8: There is a '6' in the 5th position.

Revised: April 24, 2024