Assignment 55

 "One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories." -- (Philip J. Davis, 1964) The Bernoulli Family. For 200 years, no fewer than eleven members of this remarkable family made contributions to mathematics. Four family members are mentioned here: Nicholas I (1695-1726) developed properties of curves, differential equations, and calculus. Daniel I (1700-1782) wrote books about probability, astronomy, physics, and hydrodynamics. Johann II (1710-1790) wrote about the mathematical theory of heat and light. Johann III (1744-1807) wrote many papers on astronomy, the doctrine of chance, recurring decimals, and indeterminate equations. The mathematical contributions of this amazing family begin with Jacob I, who was born in 1654, and ended with Johann Gustav, who died in 1863. The Bernoullis were prolific writers, and their ideas filled the pages of many books. What does Herkimer call a lazy butcher? Answer: A meat loafer. Herky's friends: TRUDY AGES...she really liked to study history. CARY A. LOAD...this guy is a truck driver. ASSIGNMENT #55 Reading: Review Section 7.7, as necessary. Written: Find the standard deviation for these three sets of numbers:Set 1: 0,0,0,0,100,100,100,100Set 2: 30,40,45,50,50,55,60,70Set 3: 50,50,50,50,50,50,50,50 On page 450, do problems 36, 37, 38.
 Mathematical word analysis:PRIME: From the Latin word primus, which means "first." Since every interger is a prime or a product of primes, the prime numbers are the first (or most basic) numbers in multiplication.
STANDARD DEVIATION is a measure of spread. Basically, it provides some information about how numbers in a set are spread about the mean. If a standard deviation is 0, then all scores are identical.

Here is a simple example showing the computation of a standard deviation:

Consider the set 5, 7, 12. The mean is (5+7+12)/3 = 8. The standard deviation is

÷[[(5-8)2 + (7-8)2 + (12-8)2]/3] = ÷(26/3) = 2.9439.

Here is a simple number set with related statistics:

Set: 1, 1 ,2 ,3, 4, 5, 8, 8, 9, 10, 26

Mean = 77/11 = 7.
Standard deviation = 6.7555.
Modes are 1 and 8 (This set is bimodal).
Minimum = 1.
25th percentile = 2.
Median = 50th percentile = 5.
75th percentile = 9.
Maximum = 26.
Range = 26 - 1 = 25.
Interquartile range = 9 - 2 = 7.

"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write."

-H. G. WELLS

Problem: The sets A, B, and C as defined below each have a mean of 50. Find the standard deviation of each set.

 A = {50, 50, 50} B = {40, 50, 60} C = {0, 50, 100}

Standard deviation of set A = ÷[[(50-50)2 + (50-50)2 + (50-50)2]/3] = 0.

Standard deviation of set B = ÷[[(40-50)2 + (50-50)2 + (60-50)2]/3] = ÷[200]/3] = 8.16.

Standard deviation of set A = ÷[[(0-50)2 + (50-50)2 + (100-50)2]/3] = ÷[5000]/3] = 40.82.

NOTE: The above examples show that just knowing the mean of a set of numbers doesn't tell you anything about how the numbers are distributed.