Machinery's Handbook, 31st Edition
138 STATISTICAL ANALYSIS Coefficient of Variation (V): Used to evaluate or control the variability in data points. The coefficient of variation is calculated by dividing the sample standard deviation S by the sample mean X and expressing the result as a percent: (3) Absolute Error of the Sample Mean (A x ): Calculated by dividing the sample standard deviation by the square root of the number of data points. The result is expressed in the same unit of measure as the sample standard deviation and the sample mean: (4) Relative Error of the Sample Mean (R x ): Calculated by dividing the absolute error of the sample mean by the sample mean and expressing the result as a percent: (5) Critical Value of “t-Distribution” (“Student’s t-distribution”): The t - distribution was discovered in 1908, by W. S. Gosset, who wrote under the name “Student” (the brewery he worked for did not allow him to publish under his own name). The critical value of t depends on the number of degrees of freedom and the probability of error. If a 95% two- sided confidence is used for statistical analysis, then the probability of error is 5%, or 2.5% per side. A 5% probability of error provides practical accuracy, which is commonly acceptable in various engineering calculations. The t -distribution is broader than the nor- mal distribution for small sample sizes, but approaches the shape of a normal distribution as sample size increases, with the difference negligible even for moderately large sample sizes ( n > 30). For a 5% probability of error, the critical value of t -distribution can be determined from Table 1, page 140, at the intersection of the column under the heading t 0.025 and the row corresponding to the number of degrees of freedom shown in the column heading d.f. Population Mean ( m ): The normal distribution has two parameters: the population mean m and the population standard deviation S . The sample mean X is an estimate of the population mean ( X ≈ µ ), and the sample standard deviation is an estimate of the popula- tion standard deviation ( s ≈ S ). A graph of the normal distribution is symmetric about its mean m . Virtually all of the area (99.74%) under the graph is contained within the interval: µ 3 σ – ( , µ 3 σ + ) The population mean lies at the center of the normal distribution curve. Thus, almost all of the probability associated with a normal distribution falls within ± three standard deviations of the population mean m . Also, 95.44% of the area falls within ± two standard deviations of m , and 68.26% within ± one standard deviation. Confidence Interval for the Population Mean: The interval of the normal curve as- sociated with the probability of a parameter occurring is the confidence interval . The probability itself is the confidence level . The ends of this interval are the confidence limits . Larger samples tend to give better estimates of a population parameter. A higher confidence level tends to produce a wider confidence interval. Confidence levels of 90%, 95%, and 99% are commonly used. For example, a 95% confidence level implies that the true population parameter (say, a measurement being looked at) has a 95% probability of falling within the presented interval. Equations (1) through (5) describe a sample mean that is only an estimate of the true (population) mean. Therefore, it is important to define a confidence interval that deter mines a range within which the population mean lies. Such an interval depends on V S X = --- 100% A x S n = ---- R x A x X = ---(100)
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