The coefficient of range is calculated using the formula (L - S) / (L + S), where L is the largest value and S is the smallest value. Here, L = 80 and S = 54. Thus, (80 - 54) / (80 + 54) = 26 / 134, which is approximately 0.194, rounding to 0.19.
932
What is the statistical study of the spread or scatteredness of observations called?
Measures of dispersion quantify how spread out or scattered the data points are around a central value. Common examples include range, variance, and standard deviation, which collectively describe the variability within a dataset.
933
How are measures of dispersion that rely on the distance between specific values in a dataset, such as the range or interquartile range, formally classified?
Distance measures of dispersion, also known as range-based measures, quantify the spread of data by calculating the difference between two specific points in the distribution, such as the maximum and minimum (range) or the third and first quartiles (interquartile range).
934
How does the standard deviation generally compare to the mean deviation for a set of data?
The standard deviation is mathematically defined as the square root of the average of squared deviations, whereas the mean deviation is the average of absolute deviations. Because squaring values greater than one increases their magnitude, the standard deviation is typically greater than or equal to the mean deviation for any given dataset.
935
Which mathematical operations cause the value of the standard deviation to change?
The standard deviation is invariant to changes in origin (adding or subtracting a constant). However, it is affected by changes in scale (multiplying or dividing by a constant). Since option B is a valid factor that changes standard deviation, the provided answer 'None' is factually incorrect.
936
Which of the following are common real-world applications of the statistical range?
The range, defined as the difference between the maximum and minimum values, is a simple measure of dispersion. It is widely used in meteorology to track temperature variations, in manufacturing for quality control to monitor process consistency, and in finance to assess the volatility of stock prices.
937
What term describes the degree of scatter or variation of values in a dataset relative to their average?
Dispersion, also known as variability or spread, measures how far the data points in a series are spread out from the central value (the mean or median). Common measures of dispersion include the range, variance, and standard deviation, which quantify the extent to which the data deviates from the average.
938
What is the standard deviation of the first 50 natural numbers?
The variance of the first n natural numbers is given by (n^2 - 1) / 12. For n=50, the variance is (2500 - 1) / 12 = 2499 / 12 = 208.25. The standard deviation is the square root of 208.25, which is approximately 14.43. This confirms the calculation is mathematically sound.
939
Which measure of dispersion is calculated using only the two extreme values in a dataset?
The range is defined as the difference between the maximum and minimum values in a dataset. Because it only considers these two extreme observations, it is highly sensitive to outliers and does not provide information about the distribution of values between the extremes, unlike standard deviation or quartile deviation.
940
Given a sum of squared deviations of 48 and a sample size of 13, what is the sample variance?
The sample variance is calculated by dividing the sum of squared deviations by the degrees of freedom (n-1). Here, 48 divided by (13-1) equals 48/12, which results in 4.
