Numerical Descriptive Measures PDF

DSILYTC Lecture (23/01/2025) Numerical Descriptive Measures A. Measures of Central Tendency: center of a given data set; usually, it is a single value about which the observation tends to cluster (but not always) ​ Mean - sum of all observations divi...

DSILYTC Lecture (23/01/2025) Numerical Descriptive Measures A. Measures of Central Tendency: center of a given data set; usually, it is a single value about which the observation tends to cluster (but not always) ​ Mean - sum of all observations divided by total number of observations (Average) -​ It always exists for quantitative variables -​ It is unique -​ Takes into account every item of the data; it is easily affected by extreme values ​ Median - middle value of an array, denoted by Md’ -​ If n is Even: find the average of the two middle values -​ Arrange the data in array form first -​ Not easily affected by extreme values -​ Always exists and is unique ​ Mode - observations that occur most frequently in the data set ​ -​ May not be unique (can be more than one) -​ No calculations are required -​ May not exist (if all frequencies of the values are the same/equal) Considerations for choosing Measures of Central Tendency ​ For nominal variables, the mode is the only measure that can be used ​ For ordinal variables, the mode and median may be used ​ For interval - ratio variables, the mode, median, and media may all be calculated B. Measures of Position: measures that discriminate a group of scores from another group in the same data set. ​ Quantiles - divides data into an equal number of parts (1)​ Quartiles (Q): dividing data into four equal parts (2)​ Deciles (D): dividing data into ten equal parts (3)​ Percentile (P): dividing data into one hundred equal parts  Formula: (Always convert into Percentile) ​ Pk = k(n+1)/100 to locate position ​ Interpolation C. Measures of Variability: describes the extent to which the data are dispersed. -​ Variability is descriptive statistics; Describes how similar a set of scores are to each other -​ The more similar the scores = lower measure of dispersion -​ The less similar the scores = higher measure of dispersion -​ In general, the more the spread out a distribution is, the larger the measure of dispersion will be ​ Range: difference between the highest and lowest value in the data set -​ rarely used in scientific work as it is fairly insensitive ​ Variance: the mean squared the differences of the observations from their mean -​ The difference is called a deviate or deviation score (measure of dispersion), which tells us how far a given score is from the typical, or average, score. ​ Standard deviation: positive square root of the variance -​ Since squared units of measures are often awkward to deal with, the square root of the variance is often used instead. ​ Coefficient of Variation: ratio of the standard deviation to its mean expressed in percent. -​ Compare variability of two populations that are expressed in different units of measurements ​ Skewness: (Skew) measure of symmetry in the distribution of scores -​ Measure of skewness: frequency curve that is not symmetrical about the mean a.​ Positive skew: Tails off to the right (mean is greater than median) b.​ Negative skew: Tails off to the left (mean is less than median) -​ It is possible to obtain a measure of skewness which indicates both the direction and the magnitude of skewness of frequency data.  -​ If Sk < 0 = negative skew -​ If Sk > 0 = positive skew -​ If Sk = 0 - symmetrical ​ Kurtosis: measures whether the scores are spread out more or less than they would be in a normal (Gaussian) distribution. -​ Mesokurtic: K = 3 -​ Leptokurtic: K > 3 -​ Platykurtic = K < 3