# stats chp 4 - Descriptive Statistics Numerical Summaries...

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Stat 3360 1 Descriptive Statistics Numerical Summaries Chapter 4

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Stat 3360 2 Chapter 4 Numerical summaries Population versus sample Central tendency-location Variation-Spread Graphical displays of dispersion Covariance
Stat 3360 3 Descriptive Statistics - Numerical Summaries Central Tendency - location mean or average - center of gravity median - divides sample into halves mode - peaks in frequency distribution

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Stat 3360 4 Mean is the sample mean Sum all the values and divide by number values Notation - Greek symbol - μ for population data n x i Σ x
Stat 3360 5 Sample Mean Mean as a balance point. If a fulcrum is placed at the position of the mean, it will balance the histogram.

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Stat 3360 6 Caution About the Mean One should be cautious about using the mean when, There are outliers The data is skewed
Stat 3360 7 Median The median is the middle value after the data is arranged in ascending order. It is denoted by ( read x tilde). Resistant to outliers or influential observations. x ~

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Stat 3360 8 Steps for Median Rank the data in ascending order Find the middle term The value of this term is the median ~ 1 the value of the th term in a ranked data set 2 n x + ⎛⎞ = ⎜⎟ ⎝⎠
Stat 3360 9 Criteria When there is an odd number of elements it is straight forward What happens when there is an even number of elements?

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Stat 3360 10 Example 1 2321 1145 9916 First rank the data (sort ascending)
Stat 3360 11 Data Sorted 1 1 1 1 2 2 3 4 5 6 9 9 12 1 13 6.5 22 + ⎛⎞ == ⎜⎟ ⎝⎠ We are looking for the 6.5 th value And which value is that?

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Stat 3360 12 Answer Take the average of the sixth and seventh value for this dataset. ~ 23 5 2.5 22 x + ⎛⎞ == = ⎜⎟ ⎝⎠
Stat 3360 13 Mode The observation that occurs with highest frequency Works for both quantitative and qualitative data There can more than one Bimodal Multimodal

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Stat 3360 14 Relationship Between Measures of Central Tendency All are equal when distribution is symmetric Mean and mode alternate depending on direction of skewness
Stat 3360 15 Section 4.2 Variability

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Stat 3360 16 Spread Dispersion or Variation Easy measure to calculate is the range Range is difference between highest and lowest values Max minus min
Stat 3360 17 Variation Deviation Difference between observation and sample mean Note that the sum of the deviations of a sample is zero Standard deviation is square root of variance

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stats chp 4 - Descriptive Statistics Numerical Summaries...

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