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Descriptive Statistics Summarizing, Simplifying Useful for comprehending data, and thus making meaningful interpretations, particularly in medium to large data sets. Describing Useful for recognizing important characteristics of data Used in inferential statistics

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Important Characteristics of Data Center – typical data value Variation – spread in data Distribution – shape of data distribution Outliers – problems in data Time – changes over time?

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Graphical Summary Methods Pie Chart Useful for qualitative or quantitative data Bar Chart Useful for qualitative data Called a Pareto chart if bars ordered by height

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Graphical Summary Methods Frequency Histogram Useful for quantitative data A “connected bar plot” with bar height proportional to the frequency of the associated value or class (interval of values) Graphical summary of a frequency distribution (sometimes called a frequency table)

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Frequency Distribution (Table) For Discrete Data: Lists data values and corresponding counts Resulting histogram has a bar on each value with height proportional to its count For Continuous Data: Data is divided into classes (intervals of values) and the classes are listed along with the corresponding counts

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Definitions for Classes Lower Class Limit – smallest value in a class Upper Class Limit - largest value in a class Class Width – distance between consecutive lower (or upper) class limits Class Mark – midpoint of class (calculated as the mean of the lower and upper class limits) Class Boundaries – eliminates space between consecutive classes for plotting purposes

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Constructing a Frequency Table Select the class width (w) Approximated by range divided by the desired number of classes (usually between 5 and 15 in medium-sized data sets) Select lower class limit for first class Construct class limits using w as the distance between consecutive lower or upper class limits Count number of observations in each class.

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Types of Histograms Frequency – height of bar is count Relative Frequency - height of bar is relative frequency ( proportion/percentage/probability) Cumulative Frequency – height of bar is cumulative count Cumulative Relative Frequency – height of bar is cumulative relative frequency (percentile)

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Other Types of Graphs Dotplot Each value is plotted as a dot along an x-axis. Dots representing equal values are stacked. Stem-and-Leaf Plot Each value is separated into a stem (such as the leftmost value or values) and a leaf (such as the rightmost value or values) Stems are listed in order and leaves are plotted alongside the appropriate stem Ordered Stem-and-Leaf Plot

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Other Types of Graphs Scatter Diagram or Scatter Plot Plot of paired (x,y) data with x on the horizontal axis and y on the vertical axis. Useful for seeing relationship between x and y Time-Series Plot A special scatter diagram which as time plotted on the horizontal axis.

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Importance of Knowing the Distribution of Data Distribution can affect the choice of an appropriate statistic to use. Distribution can aid in determining the validity of many inferential statistics. Common data distributions Bell (normal), bi-modal, right-skewed (chi- squared, exponential), left-skewed

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Numerical Summary Methods Measures of Center (Location) The middle value or typical observation from a population. Measures of Variability The dispersion or spread in the population. Measures of Relative Standing The comparative value relative to the population.

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Measures of Center Population Mean Mean (Arithmetic Mean) The size of the population is denoted by N. The sample size is denoted by n. Sample Mean

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Measures of Center Median Middle value in the ordered data for odd n. Mean of the 2 middle values for even n. Commonly called the 50 th percentile. The location of the median in the ordered data set is:(n+1)÷2

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Measures of Center Mode Most common value (occurs most frequently) Midrange Midway between the lowest and highest value Trimmed Mean Mean of values remaining after an equal number of values are removed from each tail.

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Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median

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MeanMedianModeMidrange Trimmed Mean Values utilized? AllMiddle Most Common Extreme All but extreme Outliers? Not Robus t RobustRobust Robust Exists? Uniqu e Unique May not exist or be unique UniqueUnique Data type? Quan.Quan.AnyQuan.Quan. Compare Measures of Center

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Measures of Variation Range Distance between minimum and maximum Range = Max – Min The range does not measure the overall variability in the data. A measure is needed which incorporates the variability of every value in the data. One was is to look at deviations from the mean (x i - for each x i.

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Measures of Variation Population Variance Variance The average squared difference of the observations from the mean. Sample Variance

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Calculator Formula for Sample Variance

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Measures of Variation Population Standard Deviation Standard Deviation The square root of the average squared difference of the observations from the mean. Sample Standard Deviation

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Calculator Formula for Sample Standard Deviation

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Empirical Rule For data that is approximately bell-shaped in distribution, 68% of data values fall within 1 standard deviation of the mean, 95.4% of data values fall within 2 standard deviation of the mean, 99.7% of data values fall within 3 standard deviation of the mean,

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x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-13

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x - s x x + sx + s 68% within 1 standard deviation 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-13

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x - 2s x - s x x + 2s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% FIGURE 2-13

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x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell-shaped distributions ) 0.1% 2.4% 13.5% FIGURE 2-13

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Chebyshev’s Theorem For data from any distribution, the proportion (or fraction) of values lying within K standard deviations of the mean is always at least 1 - 1/K 2, where K is any positive number greater than 1. at least 3/4 (75%) of all values lie within 2 standard deviations of the mean. at least 8/9 (89%) of all values lie within 3 standard deviations of the mean.

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Relates the standard deviation of a data set to its mean The CV is useful for comparing relative variation between two or more sets of data Coefficient of Variation

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Measures of Relative Position Standard Score or Z-Score

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Measures of Relative Position Order Statistics The order statistics, denoted by, x (1), x (2), … x (n) are the observed data values ordered from smallest to greatest.

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Measures of Relative Position Percentile The k th percentile (P k ) separates the bottom k% of data from the top (100-k)% of data. The location of P k in the order statistics is:

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Finding the Value of the kth Percentile Sort the data. (Arrange the data in order of lowest to highest.) The value of the kth percentile is midway between the Lth value and the next value in the sorted set of data. Find P k by adding the L th value and the next value and dividing the total by 2. Start Compute L = n where n = number of values k = percentile in question ) ( k 100 Change L by rounding it up to the next larger whole number. The value of P k is the Lth value, counting from the lowest Is L a whole number ? Yes No Figure 2-15

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Measures of Relative Position Quartiles The quartiles (Q 1 =P 25, Q 2 =P 50 and Q 3 =P 75 ) separate the data into fourths. Interquartile Range (IQR) The distance between the first and third quartiles: IQR=Q 3- Q 1. The IQR is a measure of variability which is less affected by outliers than the range, variance and standard deviation.

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Box-and-Whisker Diagram (Boxplot) Graphical display of the “5 Number Summary” X (1) =Min Q 1 =P 25, Q 2 =P 50, Q 3 =P 75 X (n) =Max Inner & Outer Fences Useful for identifying potential outliers in data.

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Bell-Shaped Figure 2-17 Boxplots

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Bell-Shaped Figure 2-17 Boxplots Uniform

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Bell-ShapedSkewed Figure 2-17 Boxplots Uniform

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Percentile Rank If x=P k and k is the percentile rank of x, then k is approximately equal to:

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Exploring Measures of center: mean, median, and mode Measures of variation: Standard deviation and range Measures of relative location: order statistics, minimum, maximum, percentile Unusual values: outliers Distribution: histograms, stem-leaf plots, and boxplots

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