Chapter 1: Exploring Univariate Data Section 1.1: Types of Data  ∙ Relevance of statistics - Statistics is used to gather and analyze data for any discipline  - Statistics is used to analyze surveys ∙ What is Statistics? - Statistics is used to make intelligent decisions in a world full of  uncertainty  “A knowledge of statistics provides the necessary tool to  differentiate between sound statistics conclusions and  questionable conclusions.” - Statistics is the science of collecting, organizing, and interpreting  numerical facts which we call data ∙ What is “Data”? - Statistics is the science of collecting, organizing and interpreting  numerical facts which we call data  The facts and figures collected, analyzed, and summarized for presentation and interpretation  Amount of your last purchase at a grocery store   The number of times that you access a certain website  Your name ∙ Types of Data:  - Population Data is everything or everyone we want information  about  It is a set of data that consists of all possible values pertaining to a certain set of observations or an investigation - Sample Data is a subset of the population that we have information  from  It is just a small section of the population taken for the  purpose of investigation ∙ Examples of Types of Data - Identify the population and the sample for each of the following:  University of Houston is interested in how many students buy  used books as opposed to new ones. They randomly choose  100 students at the student center to interview. o Population – All UH Students o Sample – 100 Students Samples  An elementary school is creating a new lunch menu. They  send questionnaires to students with last names that begin  with the letter M through R.  o Population – All students at this school o Sample – Students with last names that start with M  through R∙ A variable is a characteristic of an individual that can assume more  than one value - Variables can be classified as categorical (qualitative) or  quantitative (numeric).  Categorical variables – describe qualities or characteristics  that data may have o They usually represent a “type of something” such as a  type of car.   Quantitative variables – are measurements  o These will be numeric values ∙ Quantitative variables can be classified as either discrete or  continuous - Discrete quantitative variables – a countable set of values   For example: the number of lives given in a single play of a  video game - Continuous quantitative variables – data that can take on any  values within some interval   For example: the amount of time you wait in line at the  drivers license office ∙ Example: Classify the following variables as categorical or quantitative. - If quantitative, state whether the variable is discrete or continuous.  dPolitical preference o Categorical   Number of siblings o Quantitative - Discrete  Blood type o Categorical   Height of men on a professional basketball team  o Quantitative – Continuous   Time it takes to be on hold when calling the IRS at tax time o Quantitative – Continuous  Section 1.2: Mean and Median ∙ One question we want to answer about data is about its location,  particularly the location of its center. - Mean – is denoted with the Greek letter µ when referring to the  population mean and with the symbol ´x when referring to the  sample mean   Most common measure of center  Arithmetic average  We find the mean by adding up all the values and dividing by  how many.  Where n is the size of the sample and N is the size of the  population  Symbols for mean: ´x vs µ - Median – M is the midpoint of a data set such that half of the  observations are smaller and the other half are larger  Arrange all observations in order of size, from smallest to  largest  Find the middle value of the arranged observations by  counting (n + 1)/2 from the bottom of the list o If the number of observations n is odd, the mean M is  the center observation in the ordered list.  o If the number of observations n is even, the median M is the mean of the two center observation in the ordered  list - Mode – is the numerical value that appears the most frequently   Mode is used as a description of center for categorical data  The data set can have one mode, two or more modes  A data set may not have any mode ∙ Examples:  - 1. Twelve babies spoke for the first time at the following ages (in  months):  8 9 10 11 12 13 15 15 18 20 20 26 a. What is the mean of the data? ´x = (8 + 9 + 10 + 11 + 12 + 13 + 15 + 15 + 18 + 20 + 20 +  26)/12 = 14.75 b. What is the median of the data? Median = (13 + 15)/2 = 14 c. What is the mode of the data? Bimodal modes are 15 and 20 - 2. Here are the weights (in pounds) of 20 steers on an experimental  feed diet: 174 142 131 145 175 150 176 151 110 162 133 163 135 178 178 154 166 146 156 167a. What is the mean of the data?  ´x = (174 + 142 + 131 + 145 + 175 + 150 + 176 + … + 167)/20  = 154.6 b. what is the mean of the data? 110, 131, 133, 135, 142, 145, 146, 150, 151, 154, 156, 162, 163, 166, 167, 174, 175, 176, 178, 178 Median = (154 + 156)/2 = 155 c. What is the mode of the data? Mode = 178 - 3. The test scores of a class of 20 students have a mean of 71.6 and the test scores of another class of 14 students have a mean of 78.4. Find the mean of the combined group.  Mean = sum/n Class 1: 71.6 = sum/20 sum = 20(71.6) = 1432 Class 2: 78.4 = sum/14 sum = 14(78.4) = 1097.6 Mean of combined classes = (1432 + 1097.6)/(20 + 14) = 74.4 - 4. Explain why the conclusion drawn is not valid: A businesswoman calculates that the median cost of the five  business trips that she took in a month is $600 and concludes that  the total cost must have been $3000. 1 2 3 4 5 6 $600 If $400 was mean the conclusion would be correct Section 1.3: Standard Deviation and Variance ∙ Another important question we want to answer about data is about its  spread or dispersion.  - Roughly speaking, the population standard deviation, σ, tells  the average distance that data values fall from the mean.  - The standard deviation is the square root of the population  variance, σ2.  - So, what is the variance?  - The variance is the average of the squared differences of the data  values from the mean. ∙ If N is the number of values in a population with mean μ , and xi  represents each individual value in the population, then the variance is found by: ∙ And the population standard deviation is σ = √σ2∙ Most of the time we are not working with the entire population.  - Instead, we are working with a sample.  Sample variance –   Sample standard deviation –  ∙ Example: - 1. A statistics teacher wants to decide whether or not to curve an  exam. From her class of 300 students, she chose a sample of 10  students and their grades were: 72, 88, 85, 81, 60, 54, 70, 72, 63, 43 Find the mean, variance and standard deviation for this sample. ´x = (72 + 88 + 85 + 81 + 60 + 54 + 70 + 72 + 63 + 43)/10 =  68.8 s2 = [(72 – 68.8)^2 + (88 – 68.8)^2 + (85 – 68.8)^2 + … + (43 –  68.8)^2]/(10 – 1) = ~199.7 s = √199.7 = ~14.13 - 2. Suppose the statistics teacher decides to curve the grades by  adding 10 points to each score. What is the new mean, variance and standard deviation? New mean: 78 (old mean + 10) or (68.8 + 10) New s2 = ~199.7 variance and standard deviation did not  change New s = ~14.13 By adding 10 to each data point, the spread of the data does not  change. This is variance and the standard deviation are unaffected  by adding a value to each data point.  ∙ We can see from example 2 that adding the same value to all elements does not affect the variance (or standard deviation) of a set of data.  ∙ What about multiplying? - 3. Find the variance and the standard deviation for the following set  of data (whose mean is 4.5) 3, 6, 2, 7, 4, 5 Now, multiply each value by 2. What is the new variance and the  new standard deviation? Mean(x) = 4.5 Var(x) = [(3 – 4.5)^2 + (6 – 4.5)^2 + (∙ Sometimes we want to compare the variation between two groups.  - The coefficient of variation can be used for this.  - The coefficient of variation is the ratio of the standard deviation to  the mean.  - A smaller ratio will indicate less variation in the data. ∙ Example: - 4. The following statistics were collected on two different groups of  stock prices:

Portfolio A Portfolio B Sample size 10 15 Sample mean $52.65 $49.80 Sample standard deviation $6.50 $2.95

What is the median of the data?

What is the mean of the data?

∙ What is Statistics?

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What can be said about the variability of each portfolio? A: 6.5/52.65 = 0.123 B: 2.95/49.80 = 0.0592 Smaller value in B, therefore less variation Section 1.4: Range, IQR and Finding Outliers ∙ More measures of spread (or dispersion): - Range – maximum - minimum ∙ Drawbacks of range: sensitivity to outliers - Percentiles:  25th percentile, Q1 – First Quartile, or the Lower Quartile o The data point in which is above 25% of the data  50th percentile, Median or Q2 – Second or Middle Quartile,  also the Median o The middle data point  75th percentile, Q3 – Third, or the Upper Quartile  o The data point which is above 75% of the data ∙ Interquartile Range: - The values of the minimum, Q1, Q2, Q3 and the maximum make up  what is called our five number summary.  IQR – Q3 * Q1 o The IQR represents the range of the middle 50% of the  data.   This will remove any outliers from this calculation  o Five Number Summary: Contains Minimum, First  Quartile, Median, Third Quartile, and Maximum, given in  that order.  ∙ Example: - 1. Twelve babies spoke for the first time at the following ages (in  months): 8 9 10 11 12 13 15 15 18 20 20 26 Find Q1, Q2, Q3, the range and the IQR.Range = Max – Min = 26 – 8 = 18 Q2 = (13 + 15)/2 = 14 Q1 = (10 + 11)/2 = 10.5 Q3 = (18 + 20)/2 = 19 IQR = Q3 – Q1 = 19 – 10.5 = 8.5 ∙ The IQR is used to determine data classified as outliers.  - An outlier is an observation that is “distant” from the rest of the  data.  - Outliers can occur by chance or be measurement errors so it is  important to identify them. ∙ Any point that falls outside the interval calculated by Q1- 1.5(IQR) and  Q3 + 1.5(IQR) is considered an outlier. - Q1 – 1.5(IQR) 10.5 – 1.5(8.5) = -2.25 Since this value is negative, there cannot be any outliers on the low  side of the data - Q3 + 1.5(IQR) 19 + 1.5(8.5) = 31.75 Since 31.75 is larger than our maximum, we have no outliers on the  high side of our data ∙ Example: - 2. Are there any outliers in the data set given for example 1? If so,  what are they? Q1 = 10.5 10.5 – 1.5(8.5) to 19 + 1.5(8.5) Q3 = 19 [-2.25, 31.75] IQR = 8.5 No outliers ∙ There are other percentiles as well.  - The kth percentile means that k% of the ordered data values are  at or below that data value.  - For example, if the median is 100, then 50% of the ordered data  values fall at or below 100.  - Also, (100-k)% represents the amount of ordered data that falls  above the percentile data value. ∙ If you are looking for the measurement that has a desired percentile  rank, the 100Pth percentile, is the measurement with rank (or position  in the list) of nP+0.5, where n represents the number of data values in  the sample. nP + 0.5 = rank (or position) of the Pth percentile ∙ Example: - 3. In a collection of 30 data measurements, which measurement  represents the 30th percentile? N = 30 [number of data points] 100P = 30 P = 0.30 [Percentile, given as a decimal] P = 0.30 nP + 0.5 30(0.30) + 0.5 = 9.5Between the 9th and 10th value in the ordered list between x9 and x10 The 10th item in the list of data is our 30th percentile (9.5 is rounded  up … always round up).  Make sure the list is in order! ∙ Suppose you know the position (the order) of a value and want to know what percentile it is ranked at.  - In general, if you have n data measurements, x1 represents the  100(1−0.5)/ nth percentile, 2 x represents the 100(2−0.5)/ nth  percentile, and i x represents the 100(i−0.5)/ nth percentile. [100(r – 0.5)]/n gives you the percentile r = Position (rank) ∙ Example: - 4. Using the data in example 1, determine the percentile of the 4th  order statistic (x4). Data: 8, 9, 10, 11, 12, 13, 15, 15, 18, 20, 20, 26 N = 12 [number of data points] R = 4 [position in the ordered list] [100(4 – 0.5)]/12 = 29.2 11 is at the 29.2th percentile Section 1.5: Graphs and Describing Distributions ∙ Data can be displayed using graphs and there are several types of  graphs to choose from  ∙ Some of the most common graphs used in statistics are:  - Bar graph - Pie Chart - Dot plot - Histogram - Stem and leaf plot - Box plot - Cumulative Frequency plot ∙ So how do we create these different graphs and what type of graph  would be best for our data? ∙ Graphs and Describing Distributions - Let’s start with an example: - Height measurements for a group of people were taken.   The results are recorded below (in inches): 66, 68, 63, 71, 68, 69, 65, 70, 73, 67, 62, 59, 63, 68, 71, 63, 63, 60, 64, 66, 58 - We will organize this data using different graphs:  A bar graph is created by listing the categorical data along  the x-axis and the frequencies along the y-axis.o Bars are drawn above each data value.  Each bar represents the frequency of the  individual category Chocolate: 12 Strawberry: 13 Vanilla: 10 Other: 5  A dot plot is made simply by putting dots above the values  listed on a number line. Dotchart(x)  A stem and leaf plot, the data is arranged by values.  o The digits in the largest place are referred to as the  stem and the digits in the smallest place are referred to  as the leaf (leaves).  o The leaves are displayed to the right of the stem.   A split stemplot divides up the stems into equal groups.  o Back-to-back stempots can be used when comparing  two sets of data. Stem(x) 5 | 8, 9 Line 1: 58, 59 6 | 0, 2, 3, 3, 3, 3, 4, 5, 6, 6, 7, 8, 8, 8 Line 2: 60, 62, 63, 63,  63, 63, 647 | 0, 1, 1, 3 Line 3: 65, 66, 66, 67, 68, 68, 68 Stem = tens digit Line 4: 70, 71, 71, 73 Leaf = ones digit  Histograms are created by first dividing the data into  classes, or bins, of equal width.  o Next, count the number of observations in each class.  o The horizontal axis will represent the variable values  and the vertical axis will represent your frequency or  your relative frequency. Hist(x)  Boxplots not only help identify features about our data  quickly (such as spread and location of center) but can be  very helpful when comparing data sets. o How to make a box plot:  Order the values in the data set in ascending  order (least to greatest).  Find and label the median.  Of the lower half (less than the median—do not  include), find and label Q1.  Of the upper half (greater than the median—do  not include), find and label Q3.  Label the minimum and maximum.  Draw and label the scale on an axis. Plot the five number summary.  Sketch a box starting at Q1 to Q3.  Sketch a segment within the box to represent the  median.  Connect the min and max to the box with line  segments. o Note: If data contains outliers, a box and whiskers  plot can be used instead to display the data.   In a box and whiskers plot, the outliers are  displayed with dots above the value and the  segments begin (or end) at the next data value  within the outlier interval.  A pie chart is a circular chart, divided into sectors, indicating  the proportion of each data value compared to the entire set  of values.  o Pie charts are good for categorical data.  A cumulative frequency plot of the percentages (also called an ogive) can be used to view the total number of events that  occurred up to a certain value. o Example: Here is an ogive for Hudson Auto Repair’s cost of parts sold: ∙ Patterns and shapes:- Uniform graphs - Symmetric graphs - Some other features - Bell Shaped - Skewed right - Skewed left

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