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Get Full Access to UH - MATH 2311 - Class Notes - Week 1
Get Full Access to UH - MATH 2311 - Class Notes - Week 1

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UH / Math / MATH 2311 / characteristics that start with m

Description: Covers Section 1.1 - 1.5 throughout Chapter 1 notes
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Portfolio A Portfolio B Sample size 10 15 Sample mean \$52.65 \$49.80 Sample standard deviation \$6.50 \$2.95

∙ 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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