Standard+Deviation


 * Tim, Ryan, and Dylan **


 * __ What is standard deviation? __**
 * Standard Deviation is a measure of the variability or diversity of a set of data. In other words, standard deviation is a measure of how spread out numbers are. It is used in statistics and probability theory for this reason. It is calculated by taking the square root of the variance.**


 * Low standard deviation- ** means the data points are close to the arithmetic mean
 * High standard deviation **- means the data points are spread out over a larger range
 * This means there can be multiple standard deviations, meaning multiple distances from the average.
 * To find more about means go to Mean, Median, and Mode.

__**Calculating the Standard Deviation-**__
To calculate the standard deviation of a raw set of data you must first calculate the arithmetic mean of the set. Once you have calculated the mean you can calculate the variance of the set.


 * __Key terms-__**

**Variance**- the average of the squared differences from the mean, the symbol σ 2

**Means**- more to be added...
 * mean of a set: [[image:https://lh5.googleusercontent.com/r4ww2vscOWWackIbv8Vh9nNCeyPxNxI-Z6aVD_UiYXVSudDuOToFCh8JQfdC-1l-s48Zhd4IQusbzVPpcB_SqPT7it7TJcJcAc0D8npy9KQzReQgojI]]
 * mean of a population: µ is the average of all values in a population

__**Graphical representation of calculating the Standard Deviation-**__ -**//Each dog height has a certain distance from the mean (average height).//** //**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">-The Standard deviation is the average of these distances (the first standard deviation is a distance of 147) **// //**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">-So the average distance a dog's height is from the mean is 147units. **// //**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">-Dogs that deviate too much from the mean are considered outliers. **// //**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">-This allows you to see the distribution of the Data. **//
 * //-The green line represents the mean of the set of data.//**

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> //**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;"> - **//**This graph is known as the "Gaussian Distribution Function," "Bell Curve" or "Normal Distribution curve."** <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">//**-The areas represents the % of data covered or within n-standard deviations**// <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">//**-For this set of data it takes 3 standard deviations to cover the whole set.**// <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">//**-The less standard deviations you have to go through to cover the majority of the data, the more likely the data is accurate.**//
 * __<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Higher Order Standard Deviations __<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">- **
 * -Along with Standard Deviation, this graph is also used to calculate Binomial Probability and (Probability Distribution)**
 * -The x (right graph) represents the mean, and the + and - represent up or down from the mean.**
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">-//The red part represent 1 standard deviation (1-sigma) (down from the mean, and up from the mean)// **

__Two Basic Kinds of Standard Deviation__-
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">__Population Standard Deviation__- equal to the square root of the variance
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">symbol= σ (lowercase sigma)
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">This is for when you have all of the data, meaning the population's data.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Equation:
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">__<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Sample Standard Deviation __
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">symbol= s
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">This is when you do not have all the data from the population, but only a small set of the data. (sample).
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Equation:
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The change of N in the Population Standard Deviation to N-1 in the Sample Standard Deviation is known as Bessel’s Correction. Although the reasons are beyond this text.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">This link is the proof of Bessel's Correction

__**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Analysis of the formula: **__
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Taking the square: taking the square of the distances stops the error of negative values and its opposite from canceling out. Absolute values would give an invalid answer (you can try this as an self-exercise).
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">1/N: takes the average of the square of the means: where N is the amount of data/differences from the mean.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">[[image:https://lh4.googleusercontent.com/a-jNqCKfz0aM928qPvthmtpdYm548Yj2wFJZcNCMpbM44j10tMTDi6kiSSb3qwIwzn82LCjAswZ6hwPyY8X59PPW4cDcBgXo4W_VBOhqxpnFvBoeWRo]]<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">: the difference between the data value and the arithmetic mean value.

**__Practice Problems(Non-SAT 2)-__**
//**-All questions on this page utilize µ and are to 1-sigma.**//


 * Simple Standard Deviation questions-**
 * __Instructions__- find the first order standard deviation for the following sets of data.**


 * List 1: 1,3,4,8 [|Answer, list 1] **
 * List 2: 2,3,6,9 [|Answer, list 2] **
 * List 3: 1,2,3,4 [|Answer, list 3] **
 * List 4: 3,4,5,10 [|Answer, list 4] **

These are practice problems from ([|This] website is where you can obtain the worked-out solutions to the following practice problems).

1) Consider the following three data sets A, B and C. > > B = {10,10,10,10,10} > > C = {1,1,10,19,19} > > > a) Calculate the standard deviation of each data set. > > b) Which set has the largest standard deviation?
 * A = {9,10,11,7,13}

Consider the table: 2) Calculate the standard deviation of the salaries of the 20 people.
 * ~ salary(in $) ||~ frequency ||
 * 3500 || 5 ||
 * 4000 || 8 ||
 * 4200 || 5 ||
 * 4300 || 2 ||


 * Conceptual questions-harder questions to further understand the concept of standard deviation.**

3) A given data set has a mean μ and a standard deviation σ. (Hint: everything is symbolically done, and any amount of data is sufficient: for example x+k is one value) > > b) What are the new values of the mean and the standard deviation if each data value of the set is multiplied by the same constant k?Explain.
 * a) What are the new values of the mean and the standard deviation if the same constant k is added to each data value in the given set?Explain.

4) If the standard deviation of a given data set is equal to zero, what can we say about the data values included in the given data set?

**__SAT II style questions/AP Statistics-__**

1) A data set has a standard deviation equal to 1. If each data value in the data set is multiplied by 4, then the value of the standard deviation of the new data set is equal to A) 0.25 B) 0.50 C) 1 D) 2 E) 4

2) Suppose X and Y are independent random variables. The variance of X is equal to 16; and the variance of Y is equal to 9. Let Z = X - Y.

What is the standard deviation of Z?

(A) 2.65 (B) 5.00 (C) 7.00 (D) 25.0 (E) It is not possible to answer this question, based on the information given.

=
3)For a class test, the mean score was 65, the median score was 71, and the standard deviation of the scores was 7. The teacher decided to add 5 points to each score due to a grading error. Which of the following statements must be true for the new scores? (**Note:** this question may require not knowledge of mean and median, which you can find a link to that wiki page at the top of this page) ======
 * 1) The new mean score is 70.
 * 2) The new median score is 76.
 * 3) The new standard deviation of the scores is 12.
 * || (A) || Non ||
 * || (B) || I only ||
 * || (C) || II only ||
 * || (D) || I and II only ||
 * || (E) || I, II, and III ||

__Solutions to SAT II/AP Statistics type questions__-
<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">A property of standard deviation is that if all values in a set of data are multiplied by a constant the standard deviation is multiplied by that constant.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 24px; text-decoration: none; vertical-align: baseline;">Solution- question 1 **

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">If you go to conceptual questions number two and go to the solution for that questions it is the exact same idea.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">((x+4)+(y+4)+(z+4))/3=(x+y+z)/4 +4*3/4=µ+4

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The variance = ((x+4-(µ+4))^2+(y+4-(µ+4))^2+(z+4-(µ+4))^2)/3=((x-µ)^2+(y-µ)^2+(z-µ)^2)/3, thus the variance is the same, and thus the standard deviation is the same. The answer is C 1, which is the standard deviation of the data without the constant.

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 24px; text-decoration: none; vertical-align: baseline;">Solution- question 2
<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The correct answer is B. The solution requires us to recognize that Variable Z is a combination of two independent random variables. As such, the variance of Z is equal to the variance of X plus the variance of Y.

<span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 15px; text-align: center; text-decoration: none; vertical-align: baseline;">Var(Z) = Var(X - Y) = Var(X) + Var(Y) = 16 + 9 = 25 <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The standard deviation of Z is equal to the square root of the variance. Therefore, the standard deviation is equal to the square root of 25, which is 5.


 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 24px; text-decoration: none; vertical-align: baseline;">Solution-question 3 **

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">The answer is D

<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Explanation-For this type of question you need to evaluate each statement separately. Statement I is true. If you add 5 to each number in a data set, the mean will also increase by 5. Statement II is also true. The relative position of each score will remain the same. Thus, the new median score will be equal to 5 more than the old median score. Statement III is false. Since each new score is 5 more than the old score, the spread of the scores and the position of the scores relative to the mean remain the same. Thus, the standard deviation of the new scores is the same as the standard deviation of the old scores.

**__Standard Deviation properties-__**

 * **__NOTE__**-please skip this section until you have tried the conceptual questions and the Sat II style question 1 both close to the end of the wiki, if you cannot figure it out come back here
 * __Adding a constant to all of the data__-the standard deviation does not change, in addition to the range
 * __Multiplying a constant by all of the data__- the new standard deviation is that constant times the old standard deviation, same for the range. The variance of a set of data is squared.

__**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Ways to Use Standard Deviation: **__
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Create a line of best fit on from scatter-plot ("Regression" or known as the least squares method")
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Standard deviation allows you to find the distance each data point is from the approximated best-fit line
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Through the use of calculus you can minimize these distances and find the best-fit line
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">Figure out the outliers of a set of data (Range)
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 15px; text-decoration: none; vertical-align: baseline;">revealing how consistent your results are and how far out of range certain data points are from the area where most of the data is.

__Bibliography__-

 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://hyperphysics.phy-astr.gsu.edu/hbase/math/immath/gauds.gif__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://www.shodor.org/unchem/math/stats/index.html__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://campus.udayton.edu/~physics/lhe/HELPstandard_deviation_practice.htm__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://www.analyzemath.com/statistics/mean.html__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://www.analyzemath.com/practice_tests/sat_subject/level_2_sample_1.html__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://stattrek.com/descriptive-statistics/variability.aspx__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://stattrek.com/ap-statistics/practice-test.aspx__]


 * <span style="background-color: transparent; color: #1155cc; font-family: Arial; font-size: 15px; vertical-align: baseline;">[|http://www.analyzemath.com/practice_tests/sat_subject/level_2_sample_1.html__]


 * [|http://en.wikipedia.org/wiki/Bessel%27s_correction#Proof_that_Bessel.27s_correction_yields_an_unbiased_estimator_of_the_population_variance]


 * [|http://sat.collegeboard.org/practice/practice-test-section-review?pageId=practiceSubjectTestMathLevel2&practiceTestSectionIDKey=Subject.MATH_LEVEL_2&conversationId=ConversationStateUID_1&header=Mathematics%20Level%202][|subHeader=SAT%20Subject%20Test%20in%20Mathematics%20Level%202%20Practice#subQsReviewContent]
 * ====== [] -proof that standard deviation is involved in the least squares regression.======