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Chapter 4

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Chapter 4 – Numerical Descriptive Technique Measures of Central Location - Mean, Median, Mode Measures of Variability - Range, Standard Deviation, Variance, Coefficient of Variation Measures of Relative Standing - Percentiles, Quartiles Measures of Linear Relationship - Covariance, Correlation, Determination, Least Squares Line Measures of Central Location The arithmetic mean, a.k.a. average, shortened to mean, is the most popular & useful measure of central location. How it is Computed? It is computed by simply adding up all the observations and dividing by the total number of observations: MEAN = SUM OF THE OBSERVATIONS/NUMBER OF OBSERVATIONS SIMPLE EXAMPLE Suppose observations are 5, 8, 6, 9, 10, 4 Sum of observations = 5+8+6+9+10+4 = 42 Number of observations = 6 Mean = 42 / 6 = 7 You can calculate Arithmetic Mean for Population as well as for Sample. NOTATION - When referring to the number of observations in a population (as a whole), we use uppercase letter N - When referring to the number of observations in a sample, we use lower case letter n - The arithmetic mean for a population is denoted with Greek letter “mu”: μ - The arithmetic mean for a sample is denoted with an “x-bar”: Arithmetic Mean Population – Sum all x for 1 to N i.e. 5 8 6 9 10 4 6 over Sum Xi = x1+, x2+, x3+...+x6 i=1 Properties of Median - A set of data has only one median. The median is unique - It is not affected by extremely large or small values and is therefore a valuable measure of location when such values occur. - It can be computed for interval and ordinal data. - Median = (8+9)/ 2 = 8.5 Measures of Central Location - The mode of a set of observations is the value that occurs most frequently. - A set of data may have one mode (or modal class), or two, or more modes. - Mode is a useful for all data types, though mainly used for nominal data. - For large data sets the modal class is much more relevant than a single-value mode. =MODE (range) in EXCEL Note: if you are using Excel for your data analysis and your data is multi-modal (i.e. there is more than one mode), Excel only calculates the smallest one. You will have to use other techniques (i.e. histogram) to determine if your data is bimodal, trimodal, etc. M/C – Comparing Measures of Central Tendency – positively, negatively or symmetrical Which is the best? With three measures from which to choose, which one should we use? The mean is generally our first selection. However, there are several circumstances when the median is better. The mode is seldom the best measure of central location. One advantage the median holds is that it not as sensitive to extreme values as is the mean. This value is only exceeded by only two of the ten observations in the sample, making this statistic a poor measure of central location. The median stays the same. When there is a relatively small number of extreme observations (either very small or very large, but not both), the median usually produces a better measure of the center of the data. For ordinal and nominal data the calculation of the mean is not valid. Median is appropriate for ordinal data. For nominal data, a mode calculation is useful for determining highest frequency but not “central location”. Compute the Mean to • Describe the central location of a single set of interval data Compute the Median to • Describe the central location of a single set of interval or ordinal data Compute the Mode to • Describe a single set of nominal data Geometric Mean - The arithmetic mean is the single most popular
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