Statistics Biography — Ronald A. Fisher

Sir Ronald Aylmer Fisher was a British geneticist and statistician. To create genetic experiments that yielded greater results with less effort, he pioneered the use of statistics in experimentation, and came up with the now widely used concepts of variance and randomization. Fisher wrote about 300 papers and seven books during his prodigious career.

The youngest of seven children, Ronald Fisher was born on February 17, 1890, in a northern suburb of London. Because of poor eyesight, the young Fisher was not allowed to read or write under artificial light. Consequently, he rarely took notes at lectures he attended, and he preferred to solve problems mentally rather than on paper. He developed a facility for visualizing complex geometrical relationships in his mind. This ability later proved fruitful, as his geometrical interpretation of statistics led him to previously unattainable results.

In 1909, Fisher earned a scholarship to attend Gonville and Caius College in Cambridge, where he specialized in mathematics and theoretical physics while also studying genetics. Fisher graduated from the University of Cambridge with a B.A. in astronomy in 1912. While there, he gained an interest in the theory of errors in astronomical observations, which eventually led him to a career in statistical research. For the next six years, Fisher searched for the right type of occupation, even working briefly as a farm laborer in Canada. Primarily, however, he worked as a statistician for the Mercantile and General Investment Company in London (1913 to 1915) and as a public school teacher (1915 to 1919). He was unhappy and, apparently, ineffective as a teacher–throughout his career, he was recognized as a brilliant thinker who had difficulty explaining his ideas to others. In 1917, he married Ruth Eileen Guinness; they had eight children and eventually separated.

Even though his jobs did not support research, Fisher published several papers during this period. He wrote two papers on eugenics (the science of improving the human race through selective mating); his concern that the less talented lower classes produced offspring at a faster rate than the more capable upper classes influenced his personal choice to have a large family.

In 1919, he accepted a position as statistician for the Rothamsted Agricultural Experimental Station near Harpenden, Hertfordshire. His work on plant-breeding experiments combined biology and statistics. At Rothamsted, he developed a new technique by which scientists could vary different elements in an experiment to determine the probability that those variations would yield different results. He published his findings in the book Statistical Methods for Research Workers (1925).

While at Rothamsted, Fisher also introduced new theories about randomization and variance, included in his work The Genetical Theory of Natural Selection (1930), which are now widely used in the field of genetics. His goal was to design plant-breeding experiments to yield the maximum results while using the least amount of time, effort, and money. One problem he discovered was biased selection of materials, which could lead to inaccurate results. To avoid this, Fisher introduced the concept of randomization, which provided that experiments must be conducted among a random sample of the entire population, and must be repeated on a number of control subjects to ensure validity.

Fisher also introduced his concept of variance. At the time, scientists were only able to vary one factor at a time in experiments, allowing for only one potential result. He proposed instead a statistical procedure by which experiments would be designed to answer several questions at once. This was accomplished by dividing each experiment into a series of sub-experiments, each of which differed enough to provide several unique outcomes. Fisher summed up his statistical work in his definitive work, Statistical Methods and Scientific Inference (1956).

Fisher was knighted in 1952. His last years were spent conducting research in Australia, where he died on July 29, 1962.

Statistics Biography — William Sealy Gosset

William Sealy Gosset (June 13, 1876 – October 16, 1937) is famous as a statistician, best known by his pen name Student and for his work on Student’s t-distribution.

William Sealy Gosset was born in Canterbury, England, the latest descendant of an old Huguenot family that had left France after the revocation of the Edict of Nantes. He studied at Winchester, then at Oxford, where he focused on mathematics and natural sciences. Upon graduation, he joined Arthur Guinness and Son, a Dublin brewery, and he remained employed there throughout his life, ultimately becoming chief brewer at a new brewery in London.

Early on, Gosset saw a need for careful scientific analyzes of a variety of processes, from barley production to yeast fermentation, all of which profoundly affected the quality of the brewery’s final product, beer. His firm sent him to study under Karl Pearson in London. At the time, the theory of estimation based on large samples had been fully worked out, but Gosset noticed a void with respect to small-sample estimation theory. Small samples, however, were typical of Gosset’s work at the brewery. So Gosset developed the theory himself. In a now famous 1908 paper, The Probable Error of a Mean, he noted that s is an erratic estimator of s when n is small; hence, customary measures of the precision of estimates were invalid for small samples and unknown s. His paper presented the sampling distribution of a statistic now known as Student’s t and introduced small-sample estimation by means of the t-distribution family. It is difficult to overestimate the importance of this achievement. It has proven fundamental to statistical inference as it exists today, not only in the realm of estimation, but also in hypothesis testing and the analysis of variance. Sir Ronald Fisher, who held great admiration for Gosset and shared with him a keen interest in agricultural experimentation, quite aptly called Gosset “the Faraday of statistics,” for Gosset had a similar ability to grasp general principles and develop them further by applying them to practical ends.

All but one of Gosset’s numerous papers were published under the pseudonym “Student” to protect the advances in his firm’s quality control from nosy competitors. For many years, an air of romanticism surrounded the appearance of “Student’s” papers, and only a few individuals knew his real identity, even for some time after his death.

Statistics Notes — Confidence intervals for Variance and Standard Deviation

Variance is a statistical measure that tells us how measured data vary from the average of the set of data.

Standard deviation is a measure of the variability or dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the average, while high standard deviation indicates that the data are “spread out” over a large range of values.

In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions.

In manufacturing, it is necessary to control the amount that a process varies. For instance, a cabinet shop will make hundred of similar cuts in the manufacturing process. It is important that the lengths of the cuts vary little or not at all to insure the construction in beautiful sets of cabinets.

Degrees of freedom is used to describe the number of values in the final calculation of a statistic that are free to vary.

Chi-square distribution is used to measure the amount of variation that exists in a population.

The point estimate for the population’s variance σ² is a sample’s variance s² .

The point estimate for the population’s standard deviation σ is a sample’s standard deviation s.

Chi-square distribution is a family of curves that are skewed positively. Each uses a degree of freedom that is equal to one less than the size of the sample. d.f. = n – 1

Degrees of freedom is used to describe the number of values in the final calculation of a statistic that are free to vary.

The area under each curve equals one.

There are two critical values for each level of confidence. The value χ²_R represents the right-tail critical value and χ²_L represents the left-tail critical value.

The table gives areas that represent the regions under the chi-square curve to the RIGHT of the critical value. This means these values are right tail values not at all like the normal distribution z table we used earlier.

Example. Find the critical values χ²_R and χ²_L for a 95% confidence interval when the sample size is 15.

Because the sample size is 15, there are 14 degrees of freedom.

The area to the right of χ²_R

The area to the left of χ²_L

Using 14 degrees of freedom and the areas 0.025 and 0.975, you can find the critical values.

From the table, you can see that χ²_R = 26.119 and χ²_L = 5.629.

.

We will use these critical values χ²_R and χ²_L to construct a confidence interval for a population’s variance and standard deviation.

A c-confidence interval for a population’s variance and standard deviation are determine from the following intervals:

for a population’s variance

for a population’s standard deviation

The probability that the confidence interval contains σ² or σ is c.

Guidelines

1. Verify that the population is normally distributed.

2. Identify the degree of freedom.

3. Find the point estimate s².

4. Find the critical values χ²_R and χ²_L that correspond to the given level of confidence.

5. Find the left and right endpoints and form the confidence interval for the population’s variance and population’s standard deviation.

Example. Sorry kiddies, but I’m not aware of a quick calculator menu interval thingie that will do this. So total pencil, paper. We’ll still use the calculator for our figures.

You randomly select 30 pieces of trim that has been cut in a cabinet shop. The sample standard deviation is 1.2 millimeters. Assuming the lengths are normally distributed, construct 99% confidence intervals for the population variance and standard deviation.

Critical values…

n = 30

d.f. = 29

Area to right = 0.005

Area to left = 0.995

From the table, χ²_R = 52.336 and χ²_L = 13.121.

Using these critical values and s = 1.2, here’s how we find the confidence interval for σ² .

The confidence interval for σ² is (0.7979, 3.1826)

The confidence interval for σ is = (0.8933, 1.7840).

So with 99% confidence, we can say that the interval (0.7979, 3.1826) contains the population’s variance. And with the same 99% confidence, we can say that the interval (0.8933, 1.7840) contains the population’s standard deviation.

Computer — The Internet Clipboard

Here’s something for you. Its an internet clipboard. You might be somewhere, no flash drive, no printer. Here’s a solution

CL1P.net – The Internet Clipboard

If you use it, let me know how well it worked.

Statistics Assignment — Handout 6320

Geometry Notes — Circles

A circle is an important shape in the field of geometry.

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the center.

The common distance of the points of a circle from its center is called its radius. A diameter is a line segment whose endpoints lie on the circle and which passes through the center of the circle. The length of a diameter is twice the length of the radius.

Circles are simple closed curves which divide the plane into two regions, an interior and an exterior.

Circle – the set of points in a plane that are an equal distance from a fixed point.

The center of the circle is the fixed point. The center of the circle is a point in the interior of the circle. It is not part of the circle. The center of the circle is equidistant from all points on the circle.

A circle is named for its center. The above circle would be called simply “Circle A”

The radius of a circle is the distance from the center of a circle to any point of the circle. The radius can also be defined as a segment from the center of a circle to a point on the circle. It is not part of the circle.

All radii of the same circle are equal in length.

Congruent circles – circles with congruent radii

The name of this circle is Circle E.

The center of this circle is point E.

EA, EB, and EC are all radii of Circle E.

EA, EB, and EC are all congruent.

Chord – a segment that joins two points on a circle. BC and FH are chords of Circle E.

Diameter – a chord that passes through the center of the circle. The diameter is also the distance across a circle. The diameter is twice the length of a radius. BC is a diameter.

Secant – a line that contains a chord. Line or we can call it line is a secant, it contains chord segment MT .

Tangent – a line in the plane of the circle that intersects the circle in exactly one point on the circle.

That point is called the point of tangency. Line is tangent to circle J at point C.

The name of this circle is Circle F

The center is Point F

The radii are FC, FE, FA, FD

AE is a diameter

DH is a chord

is a secant

is a secant

is a tangent

Point A is the point of tangency.

Points A, C, D, E, and H are on the circle. The distance from point F to any of these points is equal.

Points J and M is in the interior of circle F. It is not part of the circle. The distance from F to these points will be less than the length of a radius. Point F is also in the interior.

Points B, K, R, and T are in the exterior of circle F. The distance from point F to either of these points will be more than the length of a radius.

Concentric circles – circles in the same plane with the same center.

Sphere – the set of points that are an equal distance from a fixed point.

I love circle art.

Pre-Calculus Assignment — Book 6.3

Statistics Assignment — Handout 6310

Computer Software — KwikTrigII

I found a small program for solving triangle trigonometry problems. It works for right and oblique triangles. I prefer and expect pencil and paper, but it is a clever program.

Here’s a couple of screen shot. KwikTrig2

Statistics Assignment — Handout 6220