MATH 112 Shape of the Distribution

Department of Mathematics and Natural Sciences Shape of the Distribution 1/1 Measures of Shape Skewness measures the deviation from the symmetry. SK = 3(µ median) (1) SK = 3(¯x median) s (2) Example The scores of the students in the Prelim Exam has a median of 18 and a mean of 16. What does this indicate about the shape of the distribution of the scores? Math031 - Mathematics in the Modern World 2 Boxplot A boxplot is also called a box - and - whisker plot. It is a graphical representation of a summary of five important values; the minimum value, first quartile, the median, the third quartile, and the maximum value. The five important values are also called five - number summary of a data set. It can also be used to detect outliers. Math031 - Mathematics in the Modern World 3 Steps in constructing a boxplot Determine the five-number summary and the interquartile range. Then compute the values of the fences. The values for the fences are given below: Inner Fence: Q1 - 1.5IQR and Q3 + 1.5IQR Outer Fence: Q1 - 3IQR and Q3 + 3IQR Draw a box with the ends of the box at the first and third quartiles. Math031 - Mathematics in the Modern World 4 Draw a vertical line inside the box a the location of the median Draw horizontal dashed lines (called whiskers) from the ends of the box to the minimum and maximum values in the data set Construct fences Math031 - Mathematics in the Modern World 5 Example Construct a boxplot for the given data set: 12, 12, 13, 14, 14, 16, 17, 19, 19, 25 Math031 - Mathematics in the Modern World 6 The Normal Distribution The normal (or Gaussian) distribution or curve is defined as follows: f (x) = 1 p2⇡ e 1 2 (xµ) 2 2 where µ 0 and 0 are arbitrary constants. Denote normal distribution with mean µ and variance 2 by N(µ, 2). Math031 - Mathematics in the Modern World 7 Properties of a normal curve: It is symmetrical about the mean. The mean is equal to the median, which is also equal to the mode. The tails or ends are asymptotic relative to the horizontal line. The total area under the normal curve is equal to 1 or 100%. The normal curve area may be subdivided into at least three standard scores each to the left and to the right of the vertical axis.