what is the formula to calculate 2 sd

Written by Madelon F. Zady.

Don't be defenseless in your skivvies when you talk about CV'southward, or misfile STD's with SD's. Do you know what they mean when they talk about mean? These are the staff of life and butter statistical calculations. Make certain you're doing them correct.

EdD Assistant Professor
Clinical Laboratory Science Programme Academy of Louisville
Louisville, Kentucky
June 1999

• Mean or average
• Standard deviation
• Degrees of freedom
• Variance
• Normal distribution
• Coefficient of variation
• Alternating formulae
• References
• Self-assessment exercises
• Nearly the Writer

Many of the terms covered in this lesson are too found in the lessons on Basic QC Practices, which announced on this website. It is highly recommended that you study these lessons online or in difficult copy[i]. The importance of this electric current lesson, however, resides in the process. The lesson sets upward a design to be followed in future lessons.

Hateful or average

The simplest statistic is the mean or boilerplate. Years agone, when laboratories were beginning to assay controls, it was piece of cake to summate a mean and utilize that value as the "target" to be achieved. For example, given the post-obit ten analyses of a control cloth - xc, 91, 89, 84, 88, 93, eighty, 90, 85, 87 - the mean or Xbar is 877/10 or 87.vii. [The term Xbar refers to a symbol having a line or bar over the X,, however, nosotros will apply the term instead of the symbol in the text of these lessons because information technology is easier to present.]

The mean value characterizes the "key trend" or "location" of the data. Although the mean is the value most likely to be observed, many of the bodily values are different than the mean. When assaying control materials, it is obvious that technologists will not reach the mean value each and every fourth dimension a control is analyzed. The values observed volition evidence a dispersion or distribution near the hateful, and this distribution needs to be characterized to prepare a range of acceptable control values.

Standard deviation

The dispersion of values near the mean is anticipated and tin be characterized mathematically through a series of manipulations, as illustrated below, where the individual x-values are shown in cavalcade A.

Column A	Column B	Cavalcade C
Ten value	Ten value-Xbar	(X-Xbar)2
90	90 - 87.seven = 2.30	(two.thirty)2 = five.29
91	91 - 87.vii = 3.30	(3.30)2 = 10.89
89	89 - 87.7 = one.30	(1.thirty)two = 1.69
84	84 - 87.7 = -3.seventy	(-3.70)two = 13.69
88	88 - 87.seven = 0.xxx	(0.30)ii = 0.09
93	93 - 87.7 = 5.30	(5.30)two = 28.09
80	lxxx - 87.seven = -7.seventy	(-7.70)ii = 59.29
90	90 - 87.seven = 2.30	(2.30)2 = 5.29
85	85 - 87.7 = -2.70	(-ii.70)2 = 7.29
87	87 - 87.7 = -0.70	(-0.70)2 = 0.49
X = 877	(Ten-Xbar) = 0	(10-Xbar)² = 132.10

• The first mathematical manipulation is to sum () the individual points and summate the hateful or average, which is 877 divided by 10, or 87.7 in this case.
• The second manipulation is to subtract the mean value from each control value, every bit shown in cavalcade B. This term, shown as X value - Xbar, is chosen the difference score. As tin be seen here, individual divergence scores can be positive or negative and the sum of the deviation scores is ever zero.
• The third manipulation is to square the difference score to make all the terms positive, as shown in Cavalcade C.
• Next the squared departure scores are summed.
• Finally, the predictable dispersion or standard deviation (SD or southward) can be calculated every bit follows:

= [132.x/(x-one)]ane/2 = 3.83

Degrees of freedom

The "north-1" term in the above expression represents the degrees of freedom (df). Loosely interpreted, the term "degrees of freedom" indicates how much freedom or independence there is inside a group of numbers. For example, if yous were to sum 4 numbers to become a total, you have the freedom to select any numbers you similar. However, if the sum of the iv numbers is stipulated to be 92, the choice of the offset 3 numbers is fairly free (as long equally they are low numbers), but the last choice is restricted past the condition that the sum must equal 92. For instance, if the get-go 3 numbers chosen at random are 28, 18, and 36, these numbers add upwardly to 82, which is ten short of the goal. For the last number there is no liberty of choice. The number 10 must be selected to make the sum come out to 92. Therefore, the degrees of freedom have been limited by one and only n-i degrees of freedom remain. In the SD formula, the degrees of freedom are n minus 1 because the mean of the information has already been calculated (which imposes one condition or brake on the data fix).

Variance

Some other statistical term that is related to the distribution is the variance, which is the standard deviation squared (variance = SD² ). The SD may be either positive or negative in value because it is calculated every bit a foursquare root, which can be either positive or negative. By squaring the SD, the problem of signs is eliminated. 1 common application of the variance is its use in the F-test to compare the variance of 2 methods and determine whether in that location is a statistically significant deviation in the imprecision between the methods.

In many applications, however, the SD is often preferred considering it is expressed in the same concentration units as the data. Using the SD, it is possible to predict the range of control values that should be observed if the method remains stable. As discussed in an earlier lesson, laboratorians oftentimes utilize the SD to impose "gates" on the expected normal distribution of command values.

Normal or Gaussian distribution

Traditionally, afterward the discussion of the mean, standard difference, degrees of liberty, and variance, the next pace was to describe the normal distribution (a frequency polygon) in terms of the standard deviation "gates." The figure here is a representation of the frequency distribution of a large set of laboratory values obtained by measuring a single control material. This distribution shows the shape of a normal curve. Note that a "gate" consisting of ±1SD accounts for 68% of the distribution or 68% of the expanse under the curve, ±2SD accounts for 95% and ±3SD accounts for >99%. At ±2SD, 95% of the distribution is inside the "gates," 2.5% of the distribution is in the lower or left tail, and the same amount (2.v%) is present in the upper tail. Some authors call this polygon an error bend to illustrate that small errors from the mean occur more frequently than big ones. Other authors refer to this bend as a probability distribution.

Coefficient of variation

Another way to describe the variation of a test is calculate the coefficient of variation, or CV. The CV expresses the variation as a percentage of the hateful, and is calculated as follows:

CV% = (SD/Xbar)100

In the laboratory, the CV is preferred when the SD increases in proportion to concentration. For example, the data from a replication experiment may show an SD of iv units at a concentration of 100 units and an SD of viii units at a concentration of 200 units. The CVs are 4.0% at both levels and the CV is more useful than the SD for describing method operation at concentrations in betwixt. However, not all tests volition demonstrate imprecision that is constant in terms of CV. For some tests, the SD may exist abiding over the belittling range.

The CV likewise provides a general "feeling" about the performance of a method. CVs of 5% or less generally give us a feeling of expert method performance, whereas CVs of x% and college audio bad. However, you should look carefully at the mean value before judging a CV. At very low concentrations, the CV may be high and at high concentrations the CV may be low. For example, a bilirubin test with an SD of 0.1 mg/dL at a hateful value of 0.five mg/dL has a CV of 20%, whereas an SD of 1.0 mg/dL at a concentration of 20 mg/dL corresponds to a CV of 5.0%.

Alternate formulae

The lessons on Basic QC Practices cover these same terms (meet QC - The data calculations), but employ a different form of the equation for calculating cumulative or lot-to-date means and SDs. Guidelines in the literature recommend that cumulative means and SDs be used in computing command limits [2-4], therefore it is important to be able to perform these calculations.

The cumulative mean can be expressed as Xbar = (x_i)_t /n_t, which appears similar to the prior hateful term except for the "t" subscripts, which refer to information from different time periods. The thought is to add together the x_i and due north terms from groups of data in order to calculate the mean of the combined groups.
The cumulative or lot-to-appointment standard departure can exist expressed as follows:

This equation looks quite unlike from the prior equation in this lesson, but in reality, it is equivalent. The cumulative standard deviation formula is derived from an SD formula chosen the Raw Score Formula. Instead of first calculating the mean or Xbar, the Raw Score Formula calculates Xbar within the square root sign.

Oftentimes in reading about statistics, an unfamiliar formula may be presented. You should realize that the mathematics in statistics is often redundant. Each procedure builds upon the previous procedure. Formulae that seem to be dissimilar are derived from mathematical manipulations of standard expressions with which yous are oft already acquainted.

References

1. Westgard JO, Barry, PL, Quam EF. Basic QC practices: Grooming in statistical quality control for healthcare laboratories. Madison, WI: Westgard Quality Corporation, 1998.
2. Westgard JO, Barry PL, Chase MR, Groth, T. A multirule Shewhart chart for quality control in clinical chemistry. Clin Chem 1981;27:493-501.
3. Westgard JO, Klee GG. Quality Management. Chapter 17 in Tietz Textbook of Clinical Chemistry, third ed., Burtis and Ashwood, eds. Philadelphia, PA: Saunders, 1999.
4. NCCLS C24-A2 certificate. Statistical quality control for quantitative measurements: Principles and definitions. National Committee for Clinical Laboratory Standards, Wayne PA, 1999.

Cocky-cess exercises

1. Manually summate the mean, SD, and CV for the following data: 44, 47, 48, 43, 48.
2. Employ the SD Calculator to calculate the hateful, SD, and CV for the following data: 203, 202, 204, 201, 197, 200, 198, 196, 206, 198, 196, 192, 205, 190, 207, 198, 201, 195, 209, 186.
3. If the information above were for a cholesterol control material, summate the control limits that would contain 95% of the expected values.
4. If control limits (or SD "gates") were ready every bit the mean +/- ii.5 SD, what percentage of the control values are expected to exceed these limits? [Hint: you need to find a table of areas under a normal curve.]
5. Describe how to calculate cumulative control limits.
6. (Optional) Show the equivalence of the regular SD formula and the Raw Score formula. [Hint: start with the regular formula, substitute a summation term for Xbar, multiply both sides by n/n, and so rearrange.]

About the author: Madelon F. Zady

Madelon F. Zady is an Assistant Professor at the University of Louisville, School of Allied Wellness Sciences Clinical Laboratory Science program and has over 30 years experience in pedagogy. She holds BS, MAT and EdD degrees from the University of Louisville, has taken other avant-garde form work from the School of Medicine and School of Teaching, and as well avant-garde courses in statistics. She is a registered MT(ASCP) and a credentialed CLS(NCA) and has worked role-time as a bench technologist for 14 years. She is a fellow member of the: American Guild for Clinical Laboratory Science, Kentucky Country Society for Clinical Laboratory Science, American Educational Research Clan, and the National Science Teachers Clan. Her teaching areas are clinical chemistry and statistics. Her research areas are metacognition and learning theory.

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