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Average (The Means)
The average of a list of numbers equals their sum divided by ho many there are.
example: 9,1,2,2,0 (Has 5 enteries)
= 2.8 (average)
Average and Histogram
A histogram balance when supported at the average (Think of Seesaw)
In a cross sectional study different subjects are compared to each other at one point in time.
Example: Many people weight are taken at one point in time. (results are compared among subjects)
In a longitudina study, subjects are followed over time, and compared with themselves at different points in time.
Example: Takes a subjects weight and Height at 20 and then again at 40 (same person)
The median of a histogram is the value with half the "AREA" (not to be confused with percentage) to the left and half to the right
Definition: Half or more of the entries are at the median or bigger and half or more are at the median or smaller
Better Defined as the middle of all entries.
Medians and Histogram tails
Long right hand tail = Average is bigger than median
Symmetric = Average is about the same as the median
long left hand tail = Average is smaller than median
The phrase "root-means-square" says how to do the arithmetic, provided you remember to read it backwards:
- 1. Square all the entries getting rid of the signs
- 2. Take the MEAN (average) of the squares)
- 3. Take the square root of the mean.
1. First Square all the entries getting rid of the signs
1,-2,-5,7 = 12
2. Take the MEAN (average of the squares).
3. Take the square root of the means
Example: Square root of 22.5 = 4.77
Standard Deviations OR (SD)
The SD measures the size of deviations for the average;
It's a deviation from the average
The SD says how far away numbers on a list are form their average. Most entries on the list will be somewhere around one SD away from the average very few will be more than two or three SDs away.
Standard Deviation Rule
68% will roughly be within 1 SD of the Average. The other 32% ware further away.
Furthermore, 95% are within 2 SDs of the average, the other 5% are further away
The SD is the r.m.s. not of the original numbers on the list, but of their deviations from the average
True for many lists, but not all.
SD=R.M.S. deviation from the average
Example: SD of 20,10,15,15
= Average of (15)
Next subtract Average from each entry:
- (20-15)+(10-15)+(15-15)+(15-15) =5 -5 0 0
- Next take R.M.S. of entries
- = =12.5
Take square root of 12.5 = 3.5
If the Average equals a negative number (-2) than you would take the deviation from the entries (meaning how far the number is from the other.
Example 4, -8, -9, 10, -3
4-(-2) = 6 [-2_____0_________4] (it is 6 numbers apart)
-8 - (-2) = -6 [-8____________-2]
-9 - (-2) = -7 [-9____________-2]
10 - (-2) = 12 [-2____0________10]
-3 - (-2) = -1 [-3__-2]
SD and R.M.S
SD is always smaller than R.M.S.
1357/4 = 4 R.M.S = 42 = 16 square root = 4
-3 -1 1 3 = 9 1 1 3/4 = (14/4 square root) = 1.87