Distribution Shapes PDF

DISTRIBUTION SHAPES  Shape of the Distribution The shape of the distribution provides information about the central tendency and variability of measurements. Three common shapes of distributions are: Normal: bell-shaped curve; symmetrical Skewed: non-normal; non-symmetrical; can...

DISTRIBUTION SHAPES  Shape of the Distribution The shape of the distribution provides information about the central tendency and variability of measurements. Three common shapes of distributions are: Normal: bell-shaped curve; symmetrical Skewed: non-normal; non-symmetrical; can be positively or negatively skewed Multimodal: has more than one peak (mode)  A. SYMMETRICAL DISTRIBUTIONS A distribution is symmetrical if the frequencies at the right and left tails of the distribution are (more or less) identical, so that if it is divided into two halves, each will be the mirror image of the other. Some special cases of symmetrical distributions are uniform distributions, where all values (or groups of values) occur with (approximately) equal frequency and bell-shaped distributions In a symmetrical distribution the mean, median, and mode are identical (more or less equal).  SYMMETRICAL DISTRIBUTIONS UNIFORM BELL - SHAPED 200 180 160 140 120 100 100 100 100 100 80 60 40 20 0 NORMAL OR BELL – SHAPED DISTRIBUTION 25 20 P e 15 r c e n 10 t 5 0 80 90 100 110 120 130 140 150 160 POUNDS  PROPERTIES OF THE NORMAL DISTRIBUTION 1. The mean, median, and mode are equal. 2. The normal curve is bell-shaped and is symmetric about the mean. 3. The total area under the normal curve is equal to 1. 4. The normal curve approaches, but never touches, the x- axis as it extends farther and farther away from the mean.  B. SKEWED DISTRIBUTIONS A skewed distribution is asymmetrical. The distribution has one “tail” stretching out much further than the other side of the distribution, indicating one or a few values that are noticeably larger/smaller than the majority. A distribution can be skewed either left or right, indicating the tail’s direction. 1. SKEWED TO THE RIGHT OR POSITIVELY SKEWED DISTRIBUTION  CHARACTERISTICS OF SKEWED TO THE RIGHT OR POSITIVELY SKEWED Scores are concentrated at the low end of the distribution Mean > median > mode Longer tail to the right or positive side The majority of the scores fall below the mean There are more low scores than high 2. SKEWED TO THE LEFT OR NEGATIVELY SKEWED DISTRIBUTION  CHARACTERISTICS OF SKEWED TO THE LEFT OR NEGATIVELY SKEWED Scores are concentrated at the high end of the distribution Mean < median < mode longer tail to the left or negative side The majority of the scores fall above or higher than the mean More high scores than low Mean Mean Mean Mode Mode Median Mode Median Median Negatively Symmetric Positively Skewed (Not Skewed) Skewed MEASURING SKEWNESS: Skewness is the measure of Pearson’s Coefficient of the shape of a Skewness Formula: nonsymmetrical distribution Sk = mean - mode Symmetric: skewness = 0 standard deviation Positive or right skewed: skewness > 0 Sk = 3(mean – median) standard deviation Negative or left skewed: skewness < 0 PEAKEDNESS OF DISTRIBUTION (KURTOSIS) Leptokurtic: high and thin Mesokurtic: normal in shape Platykurtic: flat and spread out MEASURING KURTOSIS:  K > 3: Leptokurtic Let x1 , x2 ,...xn be n observations. Then,  K = 3: Mesokurtic n  K < 3: Platykurtic n ( xi  x ) 4 Kurtosis  i 1 2 3  n 2   ( xi  x )   i 1 