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**EXPLORING DATA WITH GRAPHS AND NUMERICAL SUMMARIES**

Math Chapter 2 EXPLORING DATA WITH GRAPHS AND NUMERICAL SUMMARIES Chapter 2

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**2.1 What Are the Types of Data?**

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Variable A variable is any characteristic that is recorded for the subjects in a study Examples: Marital status, Height, Weight, IQ A variable can be classified as either Categorical or Quantitative Discrete or Continuous

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Categorical Variable A variable is categorical if each observation belongs to one of a set of categories. Examples: Gender (Male or Female) Religion (Catholic, Jewish, …) Type of residence (Apt, Condo, …) Belief in life after death (Yes or No)

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**Quantitative Variable**

A variable is called quantitative if observations take numerical values for different magnitudes of the variable. Examples: Age Number of siblings Annual Income

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**Quantitative vs. Categorical**

Math Chapter 2 Quantitative vs. Categorical For Quantitative variables, key features are the center (a representative value) and spread (variability). For Categorical variables, a key feature is the percentage of observations in each of the categories .

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**Discrete Quantitative Variable**

A quantitative variable is discrete if its possible values form a set of separate numbers: 0,1,2,3,…. Examples: Number of pets in a household Number of children in a family Number of foreign languages spoken by an individual upload.wikimedia.org

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**Continuous Quantitative Variable**

A quantitative variable is continuous if its possible values form an interval Measurements Examples: Height/Weight Age Blood pressure

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**Proportion & Percentage (Rel. Freq.)**

Proportions and percentages are also called relative frequencies.

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Math Chapter 2 Frequency Table A frequency table is a listing of possible values for a variable, together with the number of observations or relative frequencies for each value.

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**2.2 Describe Data Using Graphical Summaries**

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**Graphs for Categorical Variables**

Use pie charts and bar graphs to summarize categorical variables Pie Chart: A circle having a “slice of pie” for each category Bar Graph: A graph that displays a vertical bar for each category wpf.amcharts.com

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**Pie Charts Summarize categorical variable**

Drawn as circle where each category is a slice The size of each slice is proportional to the percentage in that category

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**Bar Graphs Summarizes categorical variable**

Vertical bars for each category Height of each bar represents either counts or percentages Easier to compare categories with bar graph than with pie chart Called Pareto Charts when ordered from tallest to shortest

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**Graphs for Quantitative Data**

Math Chapter 2 Graphs for Quantitative Data Dot Plot: shows a dot for each observation placed above its value on a number line Stem-and-Leaf Plot: portrays the individual observations Histogram: uses bars to portray the data

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**Which Graph? Dot-plot and stem-and- leaf plot: Histogram**

Math Chapter 2 Which Graph? Dot-plot and stem-and- leaf plot: More useful for small data sets Data values are retained Histogram More useful for large data sets Most compact display More flexibility in defining intervals content.answers.com

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**Dot Plots To construct a dot plot Draw and label horizontal line**

Mark regular values Place a dot above each value on the number line Sodium in Cereals

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**Stem-and-leaf plots Summarizes quantitative variables**

Separate each observation into a stem (first part of #) and a leaf (last digit) Write each leaf to the right of its stem; order leaves if desired Sodium in Cereals

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Histograms Graph that uses bars to portray frequencies or relative frequencies of possible outcomes for a quantitative variable

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**Constructing a Histogram**

Sodium in Cereals Divide into intervals of equal width Count # of observations in each interval

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**Constructing a Histogram**

Math Chapter 2 Constructing a Histogram Label endpoints of intervals on horizontal axis Draw a bar over each value or interval with height equal to its frequency (or percentage) Label and title Sodium in Cereals

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**Interpreting Histograms**

Assess where a distribution is centered by finding the median Assess the spread of a distribution Shape of a distribution: roughly symmetric, skewed to the right, or skewed to the left Left and right sides are mirror images

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Examples of Skewness

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Math Chapter 2 Shape and Skewness Consider a data set containing IQ scores for the general public. What shape? Symmetric Skewed to the left Skewed to the right Bimodal botit.botany.wisc.edu

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Shape and Skewness Consider a data set of the scores of students on an easy exam in which most score very well but a few score poorly. What shape? Symmetric Skewed to the left Skewed to the right Bimodal

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Shape: Type of Mound

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Outlier An outlier falls far from the rest of the data

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**Time Plots Display a time series, data collected over time**

Plots observation on the vertical against time on the horizontal Points are usually connected Common patterns should be noted Time Plot from 1995 – of the # worldwide who use the Internet

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**2.3 Describe the Center of Quantitative Data**

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Mean The mean is the sum of the observations divided by the number of observations It is the center of mass

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Median Midpoint of the observations when ordered from least to greatest Order observations If the number of observations is: Odd, the median is the middle observation Even, the median is the average of the two middle observations

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**Comparing the Mean and Median**

Mean and median of a symmetric distribution are close Mean is often preferred because it uses all In a skewed distribution, the mean is farther out in the skewed tail than is the median Median is preferred because it is better representative of a typical observation

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Resistant Measures A measure is resistant if extreme observations (outliers) have little, if any, influence on its value Median is resistant to outliers Mean is not resistant to outliers

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**Mode Value that occurs most often Highest bar in the histogram**

Mode is most often used with categorical data

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**2.4 Describe the Spread of Quantitative Data**

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**Range Range = max - min The range is strongly affected by outliers.**

Math Chapter 2 Range Range = max - min The range is strongly affected by outliers.

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Math Chapter 2 Standard Deviation Each data value has an associated deviation from the mean, A deviation is positive if it falls above the mean and negative if it falls below the mean The sum of the deviations is always zero

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Math Chapter 2 Standard Deviation Standard deviation gives a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations: Find mean Find each deviation Square deviations Sum squared deviations Divide sum by n-1 Take square root

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**Standard Deviation Metabolic rates of 7 men (calories/24 hours)**

Math Chapter 2 Standard Deviation Metabolic rates of 7 men (calories/24 hours)

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**Properties of Sample Standard Deviation**

Measures spread of data Only zero when all observations are same; otherwise, s > 0 As the spread increases, s gets larger Same units as observations Not resistant Strong skewness or outliers greatly increase s

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**Empirical Rule: Magnitude of s**

Math Chapter 2 Empirical Rule: Magnitude of s

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**2.5 How Measures of Position Describe Spread**

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Percentile The pth percentile is a value such that p percent of the observations fall below or at that value

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**Finding Quartiles Splits the data into four parts**

Math Chapter 2 Finding Quartiles Splits the data into four parts Arrange data in order The median is the second quartile, Q2 Q1 is the median of the lower half of the observations Q3 is the median of the upper half of the observations

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**Measure of Spread: Quartiles**

Math Chapter 2 Measure of Spread: Quartiles Quartiles divide a ranked data set into four equal parts: 25% of the data at or below Q1 and 75% above 50% of the obs are above the median and 50% are below 75% of the data at or below Q3 and 25% above Q1= first quartile = 2.2 M = median = 3.4 We are going to start out with a very general way to describe the spread that doesn’t matter whether it is symmetric or not - quartiles. Just as the word suggests - quartiles is like quarters or quartets, it involves dividing up the distribution into 4 parts. Now, to get the median, we divided it up into two parts. To get the quartiles we do the exact same thing to the two halves. Use same rules as for median if you have even or odd number of observations. Now, what an we do with these that helps us understand the biology of these diseases? Q3= third quartile = 4.35

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**Calculating Interquartile Range**

The interquartile range is the distance between the thirdand first quartile, giving spread of middle 50% of the data: IQR = Q3 - Q1

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**Criteria for Identifying an Outlier**

An observation is a potential outlier if it falls more than 1.5 x IQR below the first or more than 1.5 x IQR above the third quartile.

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**5 Number Summary The five-number summary of a dataset consists of:**

Minimum value First Quartile Median Third Quartile Maximum value

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**Boxplot Box goes from the Q1 to Q3**

Line is drawn inside the box at the median Line goes from lower end of box to smallest observation not a potential outlier and from upper end of box to largest observation not a potential outlier Potential outliers are shown separately, often with * or +

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**Comparing Distributions**

Boxplots do not display the shape of the distribution as clearly as histograms, but are useful for making graphical comparisons of two or more distributions

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Math Chapter 2 Z-Score An observation from a bell-shaped distribution is a potential outlier if its z-score < -3 or > +3

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**2.6 How Can Graphical Summaries Be Misused?**

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**Misleading Data Displays**

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**Guidelines for Constructing Effective Graphs**

Label axes and give proper headings Vertical axis should start at zero Use bars, lines, or points Consider using separate graphs or ratios when variable values differ

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