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So What Do We Know? Variables can be classified as qualitative/categorical or quantitative. The context of the data we work with is very important. Always think about the “Five W’s”—Who, What, When, Where, Why (and How)—when examining a set of data.

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**The Three Rules of Data Analysis**

The three rules of data analysis won’t be difficult to remember: Make a picture—things may be revealed that are not obvious in the raw data. These will be things to think about. Make a picture—important features of and patterns in the data will show up. Make a picture—the best way to tell others about your data is with a well-chosen picture.

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**Qualitative Data :: Making Piles**

We can “pile” the data by counting the number of data values in each category of interest. We can organize these counts into a frequency table, which records the totals & category names. A relative frequency table is similar, but gives the percentages (instead of counts) for each category.

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**What Do Frequency Tables Tell Us?**

Frequency tables and relative frequency tables describe the distribution of a categorical variable because they name the possible categories and tell how frequently each occurs. Graphs … Pie Charts & Bar Graphs (software)

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**A contingency table allows us to look at two qualitative variables together.**

Note the totals in the margins of the table. Each set of totals gives us the marginal distribution of the respective variable.

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So What Do We Know? Qualitative variables can be summarized in frequency or relative frequency tables. Categorical variables can be displayed with bar graphs and/or pie charts. A contingency table summarizes two variables at a time. From a contingency table we can find the marginal distribution for each variable or the conditional distribution for one variable conditioned on the other variable.

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**Describing Quantitative Data**

describing a quantitative variable’s distn, make sure to always tell about L.O.S.S. !!! Location/Center/Typical Value Outliers Spread/Dispersion Shape/Distribution

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**Displaying Quantitative Data**

HISTOGRAMS First, slice up the entire span of values covered by the quantitative variable into equal-width piles called bins. The bins and the counts in each bin give the distribution of the quantitative variable. One graphical display of the distribution of a quantitative variable is called a histogram, which plots the bin counts as the heights of bars (like a bar graph). A relative frequency histogram displays the percentage of cases in each bin instead of the count.

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Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values. Stem-and-leaf displays contain all the information found in a histogram.

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**A dotplot is a simple display**

A dotplot is a simple display. It just places a dot for each case in the data.

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SHAPE Symmetric Skewed Uniform or Rectangular

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Symmetric

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**Skewed Right-Tailed … Left-Tailed**

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Uniform/Rectangular

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So What Do We Know? Quantitative variables can be displayed using histograms, dotplots, and/or stem-and-leaf displays. These displays help us to see the distributions of the variables. Consider L.O.S.S. when looking at these displays! Distributions can be classified as symmetric or skewed (look at how the tails behave with respect to the rest of the distribution).

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The mean for quantitative data is obtained by dividing the sum of all values by the number of values in the data set.

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**The following are the ages of all eight employees of a small company:**

Find the mean age of these employees.

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**Thus, the mean age of all eight employees of this company is 45**

Thus, the mean age of all eight employees of this company is years, or 45 years and 3 months.

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**The calculation of the median consists of the following two steps:**

The median is the value of the middle term in a data set that has been ranked in increasing order. The calculation of the median consists of the following two steps: Sort/Arrange the data set in increasing order Find the middle term in a data set with n values. The value of this term is the median.

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The following data give the weight lost (in pounds) by a sample of five members of a health club at the end of two months of membership: Find the median.

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**First, we rank the given data in increasing order as follows:**

Therefore, the median is The median weight loss for this sample of five members of this health club is 8 pounds.

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The median gives the “center” with half the data values to the left of the median and half to the right of the median. The advantage of using the median as a measure of central tendency is that it is less influenced by outliers & skewness. Consequently, the median is preferred over the mean as a measure of central tendency for data sets that contain outliers and/or skewness.

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**The mode is the value that occurs with the highest frequency in a data set.**

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**Range = Largest value – Smallest Value**

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**The range, like the mean has the disadvantage of being influenced by outliers.**

Its calculation is based on two values only: the largest and the smallest.

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**The standard deviation is the most used measure of dispersion.**

The value of the standard deviation tells how closely the values of a data set are clustered around the mean.

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x deviation 82 95 67 92 82 – 84 = -2 95 – 84 = +11 67 – 84 = -17 92 – 84 = +8 ∑(deviation) = 0

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**standard deviation stdev =**

sqrt (sum squared deviations divided by n-1) Example :: sqrt[( )/3] sqrt(478/3) = sqrt(159.3) = 12.62

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A numerical measure such as the mean, median, mode, range, variance, or standard deviation calculated for a population data set is called a population parameter, or simply a parameter. A summary measure calculated for a sample data set is called a sample statistic, or simply a statistic.

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**For a bell shaped distribution approximately**

68% of the observations lie within one standard deviation of the mean 95% of the observations lie within two standard deviations of the mean 99.7% of the observations lie within three standard deviations of the mean

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The age distribution of a sample of 5000 persons is bell-shaped with a mean of 40 years and a standard deviation of 12 years. Determine the approximate percentage of people who are 16 to 64 years old.

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Quartiles are three summary measures that divide a ranked data set into four equal parts. The second quartile is the same as the median of a data set. The first quartile is the value of the middle term among the observations in the lower half, and the third quartile is the value of the middle term among the observations in the upper half.

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**Each of these portions contains 25% of the observations of a data set arranged in increasing order**

Q1 Q2

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**IQR = Interquartile range = Q3 – Q1**

Calculating Interquartile Range The difference between the third and first quartiles gives the interquartile range; that is, IQR = Interquartile range = Q3 – Q1

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**The following are the ages of nine employees of an insurance company:**

Find the values of the three quartiles. Where does the age of 28 fall in relation to the ages of the employees? Find the interquartile range.

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**The following data are the incomes (in thousands of dollars) for a sample of 12 households.**

Construct a box-and-whisker plot for these data.

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**Step 1. Median = (44 + 50) / 2 = 47 Q1 = (35 + 39) / 2 = 37**

Median = ( ) / 2 = 47 Q1 = ( ) / 2 = 37 Q3 = ( ) / 2 = 61 IQR = Q3 – Q1 = 61 – 37 = 24

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Step 2. 1.5 x IQR = 1.5 x 24 = 36 Lower inner fence = Q1 – 36 = 37 – 36 = 1 Upper inner fence = Q = = 97

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**Step 3. Smallest value within the two inner fences = 29**

Largest value within the two inner fences = 72

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First quartile Third quartile Median Income

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**Smallest value within the two inner fences **

Third quartile First quartile An outlier Median Smallest value within the two inner fences Largest value within two inner fences Income

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What Can Go Wrong? Do a reality check— don’t let technology do your thinking for you. Don’t forget to sort the values before finding the median … quartiles. Don’t compute numerical summaries of a categorical variable. Watch out for multiple peaks—multiple peaks might indicate multiple groups in your data.

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What Can Go Wrong? Be aware of slightly different methods— different statistics packages and calculators may give you different answers for the same data. Beware of outliers. Make a picture (make a picture, make a picture). Be careful when comparing groups that have very different spreads.

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**So What Do We Know? We describe distributions in terms of L.O.S.S.**

For symmetric distributions, it’s safe to use the mean and standard deviation; for skewed distributions, it’s better to use the median and interquartile range. Always make a picture—don’t make judgments about which measures of center and spread to use by just looking at the data.

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