How to Use Slovin’s Formula

Andale — Mon, 14 May 2012 20:33:32 +0000

If you take a population sample, you must use a formula to figure out what sample size you need to take. Sometimes you know something about a population, which can help you determine a sample size. For example, it’s well known that IQ scores follow a normal distribution pattern. But what about if you know nothing about your population at all? That’s when you can use Slovin’s formula to figure out what sample size you need to take, which is written as n = N / (1 + Ne²) where n = Number of samples, N = Total population and e = Error tolerance

Sample question: Use Slovin’s formula to find out what sample of a population of 1,000 people you need to take for a survey on their soda preferences.
Step 1: Figure out what you want your confidence level to be. For example, you might want a confidence level of 95 percent (which will give you a margin error of 0.05), or you might need better accuracy at the 98 percent confidence level (which produces a margin of error of 0.02).
Step 2. Plug your data into the formula. In this example, we’ll use a 95 percent confidence level with a population size of 1,000.
n = N / (1 + N e²) =
1,000 / (1 + 1000 * 0.05 ²) = 285.714286
Step 3: Round your answer to a whole number (because you can’t sample a fraction of a person or thing!)
285.714286 = 286

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How to find an Interquartile Range in Statistics I

Andale — Fri, 20 Apr 2012 17:20:09 +0000

In statistics, finding the difference between the biggest and smallest values in the middle fifty percent of data–can be a tricky concept to grasp at first. However, this article breaks it down into a couple of easy steps, so you’ll have the answer in no time! If you like our easy to follow explanations of statistics, check out our easy to follow book, which has hundreds more examples, just like this one.

Probability and statistics: Interquartile range (IQR)

Step 1: Put the numbers in order
1,2,5,6,7,9,12,15,18,19,27
Step 2: Find the median (How to find a median)
1,2,5,6,7,9,12,15,18,19,27
Step 3: Place parentheses around the numbers above and below the median.
Not necessary statistically–but it makes Q1 and Q3 easier to spot.
(1,2,5,6,7),9,(12,15,18,19,27)
Step 4: Find Q1 and Q3
Q1 can be thought of as a median in the lower half of the data, and Q3 can be thought of as a median for the upper half of data.
(1,2,5,6,7), 9, ( 12,15,18,19,27). Q1=5 and Q3=18.
Step 5:Subtract Q1 from Q3 to find the interquartile range.
18-5=13.

That’s the easy way to find the interquartile range in statistics! Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.
Tips: If you want to find an IQR quickly, or if you want to check your work, check out the interquartile range calculator.
If you have a boxplot, see this article to find the interquartile range.

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What is a Coefficient of Determination?

Andale — Thu, 19 Apr 2012 21:40:27 +0000

In Statistics, the coefficient of determination, R², is a value used to help interpret and analyze how many differences in one variable can be explained by a difference in a second variable. It is related directly to the coefficient of correlation, R, a value which describes how strong of a linear relationship there is between the two variables.

Calculating the Coefficient of Determination

To determine the coefficient of determination, take the square of the coefficient of correlation. As the value of correlation is always a number ranging from -1 to 1, the value of determination will always range from 0 to 1. Since the value is a square, it will always be positive, regardless of the sign of R itself.

Meaning of the Coefficient of Determination

The determination can be thought of as a percent. Roughly speaking, it tells how many of the points of data fall within the results of the line formed by the regression equation. The higher the coefficient, the higher percentage of points the line passes through when the data points and line are plotted. If the coefficient is 0.80, then 80% of the points should fall within the regression line. Values of 1 or 0 would indicate the regression line represents all or none of the data, respectively. A higher coefficient is an indicator of a better goodness of fit for the observations.

Usefulness of R

The usefulness of R² in Statistics is its ability to determine the likelihood of future events falling within the predicted outcomes. The idea is that if more samples were added, the coefficient would show the probability of the new point falling on the line. Because it is possible to gain more samples, it is possible to test the viability of determination as a prediction tool.
Similar to correlation, it should be noted that even if there is a strong connection between the two variables, determination does not prove causality. For instance, a study on birthdays may show a large number of birthdays occur specifically within a time frame of one or two months. This does not mean that the passage of time or the change of seasons causes pregnancy.

Syntax

The coefficient of determination is often given in the syntax R²_p. The p value indicates the number of columns of data, which is useful when comparing the R² of different data sets.

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How to Find an Interquartile Range Excel 2007

Andale — Mon, 16 Apr 2012 01:37:33 +0000

The interquartile range is a measure of how disbursed your data is around a mean. It’s normally used in conjunction with the mean and median of a data set. There are several ways to find an interquartile range (the easiest of which is probably our interquartile range calculator). However, you can also calculate the IQR in Excel 2007.

Step 1: Enter your data into an Excel column. For example, place your data in cells A2 to A10.
Step 2: Click a blank cell (for example, click cell B2) and then type =QUARTILE(A2:A10,1). You’ll need to replace A2:A10 with the actual values from your data set. The “1″ in this equation (A2:A10,1) represents the first quartile.
Step 3: Click a second blank cell (for example, click cell B3) and then type =QUARTILE(A2:A10,3). Replace A2:A10 with the actual values from your data set. The “3″ in this equation (A2:A10,1) represents the third quartile.
Step 4: Click a third blank cell (for example, click cell B4) and then type =B3-B2. If your quartile functions from Step 2 and 3 are in different locations, change the cell references.
That’s it!

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What is the Best Calculator for Statistics?

Andale — Tue, 03 Apr 2012 02:03:38 +0000

A common problem many statistics student encounter is the choice of the best statistical calculator. There are many models to choose from, with each having many advanced features. As of now, the TI-83, TI-86 and TI-89 are the commonest models that are widely used by large number of students in different parts of the world. A few other models by HP, Canon and others have also gained good reputation and admiration as well.

TI-83:

TI-83

The TI-83 model is a handiwork of Texas Instruments and came as an upgraded version of the TI-82. In addition to the basic functions that every scientific calculator has, the TI-83 possesses a good number of extra features such as parametric, sequence and polar graphing modes, function graphing, statistics, algebraic and trigonometric equations and functions. It does not have many calculus functions but provisions are there to download extra applications and programs from their website.

TI-83 Plus:

In 1999, the TI-83 was enhanced with a more modernized and technologically upgraded TI-83 Plus. The TI-83 Plus includes flash memory which makes it updatable. In addition to this, it also enables larger flash applications to be manually stored on the calculators. Later in 2001, the TI-83 Plus Silver edition came onto the market that boasted of a huge amount of flash memory; almost nine times compared to TI-83 Plus and a double processing speed of the former models.

TI-86:

The TI-86 is an upgraded version of TI-85 and includes an assembly support quite similar to that of TI-83 models. The TI-86 model has a much greater memory capacity while maintaining compatibility with the TI-85 programs. This is undoubtedly one of the most stylish models with a large screen size of around 128*64 pixels supporting 21*8 characters. It has a CPU size of 6Mhz ZiLOG Z80 with 128K RAM, of which 96K are available to users. It is used by many but never enjoys the same share of popularity like that of TI-83 and TI-89 models.

TI-89:

The TI-89 is one of the most technologically advanced calculator models that is available on the market today. It is the most powerful graphic calculator that offers several innovative features that are not present in any of the previous models. In addition to this, several new applications have been included to add to its versatility and capability. With the TI-89 model, students can create animations, graphical 3D rotations and contour plotting.

TI-89

Along with the normal features, the TI-89 includes a large number of associated functions such as statistics and data plots, symbolic manipulation, text editor, split screen functions, programming capabilities, variable management and provisions to connect to some other computers or calculators. The models have large, color-coded keys which makes it extremely easy to read. It also has an I/O port with a cable and is allowed on the SAT/AP/NMSQT and PSAT tests.

TI-89 Plus:

This is a modified version of the TI-89 that has almost three times the memory present in the TI-89 model. It is a powerful tool for Calculus, Physics, Mathematics, and Chemistry. It also has a flash memory three times that of its previous models.

Recommendation

As far as statistics is concerned, the best choice seems to be the TI-83 model, though the TI-86 and TI-89 models come close.

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What is Chebyshev’s Inequality?

Andale — Sun, 25 Mar 2012 20:29:24 +0000

Chebyshev’s Inequality is a formula in probability theory that relates to the distribution of numbers in a set. The formula was originally developed by Chebyschev’s friend, Irénée-Jules Bienaymé. In layman’s terms, the formula helps determine the number of values that reside in and outside the standard deviation. The standard deviation is a statistically determined number that tells how far away values tend to be from the average of the set. Roughly two-thirds of the values should fall within one standard deviation up or down from the mean.

Chebyshev’s Inequality Definition

Pafnuty Chebyshev

Chebyschev’s Inequality formula is able to prove with little provided information the probability of outliers existing at a certain interval. Given X is a random variable, A stands for the mean of the set, K is the number of standard deviations, and Y is the value of the standard deviation, the formula reads as follows: Pr(|X-A|=>KY)<=1/K², The absolute value of the difference of X minus A is greater than or equal to the K times Y has the probability of less than or equal to one divided by K squared. You can learn how to calculate Chebyshev’s Inequality here.

Chebyshev’s Theorem Uses

The formula was used with calculus to develop the weak version of the law of large numbers. This law states that as a sample set increases in size, the closer it should be to its theoretical mean. A common example is that when rolling a six-sided die, the probable average is 3.5. A sample size of 5 rolls may result in drastically different results. If you roll the die 20 times, the average should begin approaching 3.5. As you add more and more rolls, the average should continue to near 3.5 until reaching it or becoming so close to 3.5 that they are essentially equal.

Another application is in finding the probable difference between the mean and median of a set of numbers. Using a one-sided version of Chebyshev’s Inequality theorem, also known as Cantelli’s theorem, you can prove that the absolute value of the difference between the median and the mean will always be less than or equal to the standard deviation. This is handy in determining if a median you derived is statistically possible and plausibly correct.

Visit this page to easily calculate Chebyshev’s Theorem.

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What is an Interquartile Range?

Andale — Mon, 19 Mar 2012 18:50:56 +0000

Imagine all the data in a set as points on a number line. For example, if you have 3, 7 and 28 in your set of data, imagine them as points on a number line that is centered on 0 but stretches both infinitely below zero and infinitely above zero. Once plotted on that number line, the smallest data point and the biggest data point in the set of data create the boundaries of an interval of space on the number line that contains all data points in the set. The interquartile range is the length of the middle 50% of that interval of space.

The Interquartile Range Calculator on this site can calculate that interval for you. Alternatively, you may want to calculate the IQR manually. But if you want to know that the IQR is in formal terms, the interquartile range is calculated as the difference between the third or upper quartile and the first or lower quartile. Quartile is a term used to describe how to divide the set of data into four equal portions (think quarter).

IQR Example

If you have a set containing the data points 1, 3, 5, 7, 8, 10, 11 and 13, the first quartile is 4, the second quartile is 7.5 and the third quartile is 10.5. Draw these points on a number line and you’ll see that those three numbers divide the number line in quarters from 1 to 13. As such, the interquartile range of that data set is 6.5, calculated as 10.5 minus 4. The first and third quartiles are also sometimes called the 25th and 75th percentiles because those are the equivalent figures when the data set is divided into percents rather than quarters.

Use of the Interquartile Range

Image by Jhguch at en.wikipedia

Interquartile range is used to measure how spread out the data points in a set are from the mean of the data set. The higher the interquartile range, the more spread out the data points; in contrast, the smaller the interquartile range, the more bunched up the data points are around the mean. Interquartile range is one of many measurements used to measure how spread out the data points in a data set are. It is best used with other measurements such as the median and total range to build a complete picture of a data set’s tendency to cluster around its mean.
Sources:

http://mathworld.wolfram.com/InterquartileRange.html

http://www.childrensmercy.org/stats/definitions/iqr.htm

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What is a Z-Table Used For?

Andale — Tue, 13 Mar 2012 03:07:12 +0000

The z-table is short for the Standard Normal Z-Table. The Standard Normal distribution is used in hypothesis tests, including tests on proportions and on the difference between two means. The area under the whole of a normal curve is 1, or 100 percent. The z-table helps by telling us what percentage is under the curve at any particular point.

What is a Normal Probability?

The Normal Probability Distribution is often used in statistics. You’ll often see a normal curve for salaries, grades, heights, weights and IQs.

What is a Standard Normal Probability?

Every set of data has a different set of values. For example, heights of people might range from eighteen inches to eight feet and weights can range from one pound (for a preemie) to five hundred pounds or more. Those wide ranges make it difficult to analyze data, so we “standardize” the normal curve, setting it to have a mean of zero and a standard deviation of one.

Percentages under the curve

This graph shows the standardized normal graph with the percentage of results (data) that will fall between standard deviations on that graph. For example, 68.27 percent of results will fall within one standard deviation of the mean. On this graph, it’s represented by the area between z=-1 and z=1.

The z-table

Obviously a graph can only give us so much information. The above graph can tell us the area under the curve for one (z= -1 to 1), two (z= -2 to 2) and three (z= -3 to 3) standard deviations from the mean. But what about if we want to know the area between z=-0.78 and z=0.78? Or z=-1.2 and z=0.44? That’s where the z-table comes in. It tells us the area under the standard normal curve for any value between the mean, zero and any z-value.

Why Are There Two z-tables

Simply, it’s to make life easier. Sometimes you’ll want to know the area between the mean (0) and some positive value. That’s when you’ll use the regular z-table (area to the right of z). But other times you might want to know the area in a left tail. If that’s the case, use the z-table that shows the area to the left of z.

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Central Limit Theorem: “Between” Probability

Andale — Thu, 26 Jan 2012 20:03:25 +0000

Sample problem: The there are 250 dogs at a dog show who weigh an average of 12 pounds, with a standard deviation of 8 pounds. If 4 dogs are chosen at random, what is the probability they have an average weight of greater than 8 pounds and less than 25 pounds?

Step 1:Identify the parts of the problem. Your question should state:

mean (average or μ) standard deviation (σ) population size
sample size (n)
number associated with “less than” 1
number associated with “greater than” 2

Step 2: Draw a graph. Label the center with the mean. Shade the area between 1 and 2. This step is optional, but it may help you see what you are looking for.

Step 3: Use the following formula to find the z-values.

All this formula is asking you to do is:

a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 25-12=13.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8/sqrt4=4
c) Divide your result from a by your result from b: 13/4=3.25

Step 4 Use the formula from Step 3 to find the z-values. This time, use Xbar2 from Step 1 (8).

a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 8-12=-4.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8/sqrt4=4
c) Divide your result from a by your result from b: -4/4= -1

Step 5: Look up the z-value you calculated in Step 3 in the z-table.

Z value of 3.25 corresponds to .4994

Step 6: Look up the z-value you calculated in Step 4 in the z-table.

Z value of 1 corresponds to .3413

Step 7: Add Step 5 and 6 together:

.4994 + .3413 = .8407

Step 8: Convert the decimal in Step 7 to a percentage:

.8407 = 84.07%

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How to make a frequency chart and determine frequency

Andale — Fri, 30 Dec 2011 00:51:57 +0000

If you are asked to determine a frequency in statistics, it doesn’t just mean that you should just count out the number of times something occurs.

Step 1: Make a chart for your data. For this example, let’s say you’ve been given a list of twenty blood types for incoming emergency surgery patients:

A, O, A, B, B, AB, B, B, O, A, O, O, O, AB, B, AB, AB, A, O, A

On the horizontal axis, write “frequency (#)” and “percent (%)”. On the vertical axis, write your list of items. In this example, we have four distinct blood types: A, B, AB, and O.

Step 2: Count the number of times each item appears in your data.
In this example, we have:
A appears 5 times
B appears 5 times
O appears 6 times
AB appears 4 times

Write those in the “number” column. This is your frequency.

Step 3:
Use the formula % = (f / n) × 100 to fill in the next column. In this example, n = total amount of items in your data = 20. A appears
5 times (frequency in this formula is just the number of times the item appears). So we have:

(5 / 20) × 100 = 25%

Fill in the rest of the frequency column, changing the ‘f’ for each blood type.

That’s it!

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