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Chapter Three. Numerical Descriptive Measures. Commonly used Descriptive Measures:. Measures of Central Tendency Measures of Variation Measures of Position Measures of Shape. Measures of Central Tendency. Purpose: To determine the “centre” of the data values .
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Chapter Three Numerical Descriptive Measures
Commonly used Descriptive Measures: • Measures of Central Tendency • Measures of Variation • Measures of Position • Measures of Shape
Measures of Central Tendency Purpose: To determine the “centre” of the data values.
Measures of Central Tendency Answer questions • Where is the middle of my data? {Mean, Median, Midrange} • Which data value occurs most often? {Mode}
The Mean The sample mean is denoted by x-bar The population mean is denoted by µ (mu) x = individual data values X-bar = Σx / n µ = Σx / N
Example: The following are accident data for a 5 month period: 6, 9, 7, 23, & 5
To calculate the average number of accidents per month: X-bar = Σx / n X-bar = (6 + 9 + 7 + 23 + 5) ÷ 5 X- bar = 10.0
Statistic What is the average person’s monetary value to society?
The Median is the centre value in a data set when the data are arranged from smallest to largest.
By arranging the data in an Ordered Array: 5, 6, 7, 9, & 23 With an even number of observations, the value that has an equal number of items to the right and to the left is the Median. Md = 7
To calculate the median with an even number of observations, average the two center values of the ordered set. Example: With an ordered array: 5, 6, 7, & 9 Md = ( 6 + 7 ) ÷ 2 = 6.5
If there is an odd number of observations: Md = (n + 1 ) ÷ 2 where n = # of observations
Remember: Median describes the centrally placed location of a value relative to the rest of the data.
Question Is the mean or median more sensitive to extreme values (outliers)? Explain.
The mean is affected by every value. The median is unaffected by extreme values.
The mean is pulled toward extreme values. The median does not use all data information available.
Question: When dealing with data that are likely to contain outliers (personal income, ages, or prices of houses), would the Mean or Median be preferred as the measure of central tendency? Why?
Think of the Median as providing a more “typical” or “representative” value of the situation.
The Mode(Mo) The value that occurs most frequently.
Questions? • Can there be more than one mode? • Is the mode affected by extreme values? • For continuous variables, is it possible that a mode does not exist? Explain? • Is the mode always a measure of central tendency?
Give an example of when the mode may provide more useful information than the mean or the median.
Example From a purchaser’s standpoint, the most common hat or jeans size is what you would like to know, not the average hat or jeans size.
The use of any single statistic to describe a complete distribution fails to reveal important facts.
Measures of Variation Answers the question: “How spread out are my data values?”
Consider Two Scenarios Scenario 1: Jack buys a car & pays $1000. Jill buys a car & pays $21,000. Average Price = $11,000
Scenario 2: Bob buys a car & pays $10,000. Mary buys a car & pays $12,000. Average Price = $11,000
Based on the data, both scenarios report the same “average price.”
Quiz Suppose you are a purchasing agent for a large manufacturing company. Your two suppliers fill your orders in an average of 10 days. The following histograms plot the delivery time of the two suppliers.
Do the two suppliers have the same reliability in terms of making deliveries on time?
Homogeneity: the degree of similarity within a set of data values. The mean of a homogeneous data set is far more representative of the typical value than a mean of a heterogeneous data set.
If all the data values in a sample are identical, then the mean provides perfect information, the variation is zero, and the data are perfectly homogeneous.
Variation: the tendency of data values to scatter about the mean, x-bar.
If all the data values in a sample are identical, then the mean provides perfect information, the variation is zero, and the data are perfectly homogeneous.
Commonly used Measures of Variation: • Range • Variance • Standard Deviation • Coefficient of Variation (CV)
The Range Range = H – L The value of the range is strongly influenced by an outlier in the sample data.
Variance & Standard Deviation During a five week production period, a small company produced 5,9,16,17,& 18 computers, respectfully. The average = 13 computers/wk Describe the variability in these five weeks of production.
Example: Empirical Rule A company produces a lightweight valve that is specified to weigh 1365 g. Unfortunately, because of imperfections in the manufacturing process not all of the valves produced weigh exactly 1365 grams. In fact, the weights of the valves produced are normally distributed with a mean weight of 1365 grams and a standard deviation of 294 grams.
Question? • Within what range of weights would approximately 95% of the valve weights fall? 2) Approximately 16% of the weights would be more than what value? 3) Approximately 0.15% of the weights would be less than what value?
