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statistical_analysis_spring_2008 - Statistical Analysis...

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Statistical Analysis Introduction In any experiment that a person conducts to collect data, there are multiple sources of errors. As a result, it is impossible to measure with absolute certainty the “true” value of any quantity. In order to minimize the error associated with a certain measurement, the experimenter might consider taking multiple measurements in the hope that the average value of these measurements may in fact be close to the true value of that quantity. Definitions Mean : The mean of a group of measurements is the sum of the measured values divided by the total number of measurements. n x x i i = Where: i i x = x 1 + x 2 + x 3 + ……….+ x n n = number of measurements Median : The median of a group of measurements is the middle value when the data are arranged in ascending or descending order. Standard Deviation : The standard deviation measures how closely the data are clustered about the mean. 1 ) ( 2 - - = n x x s i i s 2 = also known as the variance Coefficient of Variance (CV) = % 100 × x s n-1 = also known as the degrees of freedom Precision : Precision describes the reproducibility of measurements. In other words, precision is an indicator of how close the values of multiple measurements are to each other. Precision does not necessarily indicate how close the measured values are to the true value. So, a measurement may be highly precise, but still completely wrong compared to the actual value. Standard deviation, variance, and coefficient of variation are all indicators of the precision of a measurement. In all these quantities absolute value of the difference between the measured value and the mean is known as the deviation from the mean. x x d i i - =
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Accuracy : Accuracy is defined as the nearness to the true value. A large number of measurements are considered to be accurate if the average value of the measurement is close to the true value of the measured quantity. Significant Figures The minimum number of digits needed to write a given value in scientific notation without loss of accuracy is defined as the number of significant figures. The following rules will be applied for significant figures: 1. Zeros are significant when they occur in the middle of a number or at the end of a number on the right hand side of a decimal point. 2. If the numbers to be added or subtracted have equal numbers of digits, the answer is written to the same decimal place as in the individual numbers. 3. If the numbers being added or subtracted do not have the same number of digits, the answer is limited to the least certain of the numbers. 4. When rounding off, look at all the digits beyond the last place desired. If the number is more than halfway to the next higher digit, then we round up. If the insignificant figures were less than halfway, then we round down. If the number is exactly halfway, then we round to the nearest even digit. 5.
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statistical_analysis_spring_2008 - Statistical Analysis...

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