STAT 101 Dr Kari Lock Morgan 9612 Describing
STAT 101 Dr. Kari Lock Morgan 9/6/12 Describing Data: One Variable SECTIONS 2. 1, 2. 2, 2. 3, 2. 4 • One categorical variable (2. 1) • One quantitative variable (2. 2, 2. 3, 2. 4) Statistics: Unlocking the Power of Data Lock 5
The Big Picture Population Sampling Sample Statistical Inference Statistics: Unlocking the Power of Data Descriptive Statistics Lock 5
Descriptive Statistics � In order to make sense of data, we need ways to summarize and visualize it � Summarizing and visualizing variables and relationships between two variables is often known as descriptive statistics (also known as exploratory data analysis) � Type of summary statistics and visualization methods depend on the type of variable(s) being analyzed (categorical or quantitative) Statistics: Unlocking the Power of Data Lock 5
One Categorical Variable � A random sample of US adults in 2012 were surveyed regarding the type of cell phone owned � Android? i. Phone? Blackberry? Non- smartphone? No cell phone? Statistics: Unlocking the Power of Data Lock 5
Frequency Table • A frequency table shows the number of cases that fall in each category: Android i. Phone Blackberry Non Smartphone No cell phone Total 458 437 141 924 293 2253 R: table(x) Statistics: Unlocking the Power of Data Lock 5
Proportion � Statistics: Unlocking the Power of Data Lock 5
Proportion �What proportion of adults sampled do not own a cell phone? Android i. Phone Blackberry Non Smartphone No cell phone Total 458 437 141 924 293 2253 Statistics: Unlocking the Power of Data or 13% Proportions and percentages can be used interchangeably Lock 5
Relative Frequency Table � A relative frequency table shows the proportion of cases that fall in each category Android i. Phone Blackberry Non Smartphone No cell phone 0. 203 0. 194 0. 063 0. 410 0. 130 � All the numbers in a relative frequency table sum to 1 R: table(x)/length(x) Statistics: Unlocking the Power of Data Lock 5
Bar Chart/Plot/Graph � In a barplot, the height of the bar corresponds to the number of cases falling in each category R: barchart(x) Statistics: Unlocking the Power of Data Lock 5
Pie Chart � In a pie chart, the relative area of each slice of the pie corresponds to the proportion in each category R: pie(table(x)) Statistics: Unlocking the Power of Data Lock 5
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Summary: One Categorical Variable � Summary Statistics Proportion Frequency table Relative frequency table � Visualization Bar chart Pie chart Statistics: Unlocking the Power of Data Lock 5
One Quantitative Variable World gross for all 2011 Hollywood movies Hollywood. Movies 2011 More graphics on profits for Hollywood movies Statistics: Unlocking the Power of Data Lock 5
Hollywood. Movies 2011 Statistics: Unlocking the Power of Data Lock 5
Dotplot � In a dotplot, each case is represented by a dot and dots are stacked. � Easy way to see each case Statistics: Unlocking the Power of Data Lock 5
Histogram � The height of the each bar corresponds to the number of cases within that range of the variable R: hist(x) Statistics: Unlocking the Power of Data Lock 5
Histogram vs Bar Chart This is a a) b) c) d) Histogram Bar chart Other I have no idea Statistics: Unlocking the Power of Data Lock 5
Histogram vs Bar Chart This is a a) b) c) d) Histogram Bar chart Other I have no idea Statistics: Unlocking the Power of Data Lock 5
Histogram vs Bar Chart � A bar chart is for categorical data, and the x-axis has no numeric scale � A histogram is for quantitative data, and the x- axis is numeric � For a categorical variable, the number of bars equals the number of categories, and the number in each category is fixed � For a quantitative variable, the number of bars in a histogram is up to you (or your software), and the appearance can differ with different number of bars Statistics: Unlocking the Power of Data Lock 5
Shape Long right tail Symmetric Right-Skewed Statistics: Unlocking the Power of Data Left-Skewed Lock 5
Notation � The sample size, the number of cases in the sample, is denoted by n � We often let x or y stand for any variable, and x 1 , x 2 , …, xn represent the n values of the variable x � x 1 = 97. 009, x 2 = 201. 897, x 3 = 216. 196, … Statistics: Unlocking the Power of Data Lock 5
Mean � R: mean(x) Statistics: Unlocking the Power of Data Lock 5
Median The median, m, is the middle value when the data are ordered. If there an even number of values, the median is the average of the two middle values. �The median splits the data in half. R: median(x) Statistics: Unlocking the Power of Data Lock 5
Measures of Center m = 76. 66 =150. 74 Mean is “pulled” in the direction of skewness World Gross (in millions) Statistics: Unlocking the Power of Data Lock 5
Skewness and Center A distribution is left-skewed. Which measure of center would you expect to be higher? a) Mean b) Median Statistics: Unlocking the Power of Data The mean will be pulled down towards the skewness (towards the long tail). Lock 5
Outlier An outlier is an observed value that is notably distinct from the other values in a dataset. Statistics: Unlocking the Power of Data Lock 5
Outliers Transformers Harry Pirates of the Potter Caribbean World Gross (in millions) Statistics: Unlocking the Power of Data Lock 5
Resistance A statistic is resistant if it is relatively unaffected by extreme values. �The median is resistant while the mean is not. Mean Median With Harry Potter $150, 742, 300 $76, 658, 500 Without Harry Potter $141, 889, 900 $75, 009, 000 Statistics: Unlocking the Power of Data Lock 5
Outliers � When using statistics that are not resistant to outliers, stop and think about whether the outlier is a mistake � If not, you have to decide whether the outlier is part of your population of interest or not � Usually, for outliers that are not a mistake, it’s best to run the analysis twice, once with the outlier(s) and once without, to see how much the outlier(s) are affecting the results Statistics: Unlocking the Power of Data Lock 5
Standard Deviation The standard deviation for a quantitative variable measures the spread of the data �Sample standard deviation: s �Population standard deviation: (“sigma”) R: sd(x) Statistics: Unlocking the Power of Data Lock 5
Standard Deviation �The standard deviation gives a rough estimate of the typical distance of a data values from the mean �The larger the standard deviation, the more variability there is in the data and the more spread out the data are Statistics: Unlocking the Power of Data Lock 5
Standard Deviation Both of these distributions are bell-shaped Statistics: Unlocking the Power of Data Lock 5
95% Rule If a distribution of data is approximately symmetric and bell-shaped, about 95% of the data should fall within two standard deviations of the mean. �For a population, 95% of the data will be between µ – 2 and µ + 2 � Stat. Key Statistics: Unlocking the Power of Data Lock 5
The 95% Rule Statistics: Unlocking the Power of Data Lock 5
The 95% Rule The standard deviation for hours of sleep per night is closest to a) b) c) d) e) Statistics: Unlocking the Power of Data ½ 1 2 4 I have no idea Lock 5
z-score � Statistics: Unlocking the Power of Data Lock 5
z-score � A z-score puts values on a common scale � A z-score is the number of standard deviations a value falls from the mean � 95% of all z-scores fall between what two values? -2 and 2 � z-scores beyond -2 or 2 can be considered extreme Statistics: Unlocking the Power of Data Lock 5
z-score Which is better, an ACT score of 28 or a combined SAT score of 2100? � ACT: = 21, = 5 � SAT: = 1500, = 325 � Assume ACT and SAT scores have approximately bell-shaped distributions a) b) c) ACT score of 28 SAT score of 2100 I don’t know Statistics: Unlocking the Power of Data Lock 5
Other Measures of Location Maximum = largest data value Minimum = smallest data value Quartiles: Q 1 = median of the values below m. Q 3 = median of the values above m. Statistics: Unlocking the Power of Data Lock 5
Five Number Summary � Five Number Summary: Min Q 1 25% m Q 3 25% Max 25% R: summary(x) Statistics: Unlocking the Power of Data Lock 5
Five Number Summary > summary(study_hours) Min. 1 st Qu. 2. 00 10. 00 Median 15. 00 3 rd Qu. 20. 00 Max. 69. 00 The distribution of number of hours spent studying each week is a) b) c) d) Symmetric Right-skewed Left-skewed Impossible to tell Statistics: Unlocking the Power of Data Lock 5
Percentile The Pth percentile is the value which is greater than P% of the data �We already used z-scores to determine whether an SAT score of 2100 or an ACT score of 28 is better �We could also have used percentiles: ACT score of 28: 91 st percentile SAT score of 2100: 97 th percentile Statistics: Unlocking the Power of Data Lock 5
Five Number Summary � Five Number Summary: Min Q 1 25% 0 th percentile m 25% 25 th percentile 25% 50 th percentile Statistics: Unlocking the Power of Data Q 3 Max 25% 75 th percentile 100 th percentile Lock 5
Measures of Spread �Range = Max – Min �Interquartile Range (IQR) = Q 3 – Q 1 � Is the range resistant to outliers? a) Yes The range depends entirely on the most extreme values. b) No � Is the IQR resistant to outliers? a) Yes The IQR is based off the middle 50% of the data, which will not b) No contain outliers. Statistics: Unlocking the Power of Data Lock 5
Comparing Statistics �Measures of Center: Mean (not resistant) Median (resistant) �Measures of Spread: Standard deviation (not resistant) IQR (resistant) Range (not resistant) �Most often, we use the mean and the standard deviation, because they are calculated based on all the data values, so use all the available information Statistics: Unlocking the Power of Data Lock 5
Outliers �Outliers can be informally identified by looking at a plot, but one rule of thumb for identifying outliers is data values more than 1. 5 IQRs beyond the quartiles �A data value is an outlier if it is Smaller than Q 1 – 1. 5(IQR) or Larger than Q 3 + 1. 5(IQR) Statistics: Unlocking the Power of Data Lock 5
Boxplot Outliers • Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier Q 3 Q 1 R: boxplot(x) Statistics: Unlocking the Power of Data Median Lock 5
Boxplot Which boxplot goes with the histogram of waiting times for the bus? (a) (b) (c) The data do not show any low outliers. Statistics: Unlocking the Power of Data Lock 5
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Summary: One Quantitative Variable � Summary Statistics Center: mean, median Spread: standard deviation, range, IQR Percentiles 5 number summary � Visualization Dotplot Histogram Boxplot � Other concepts Shape: symmetric, skewed, bell-shaped Outliers, resistance z-scores Statistics: Unlocking the Power of Data Lock 5
To Do �Read Sections 2. 1, 2. 2, 2. 3, 2. 4 �Do Homework 1 (due Tuesday, 9/11) �If you haven’t already… Get the textbook (at bookstore now) Get a clicker and register it (due Tuesday, 9/11) Statistics: Unlocking the Power of Data Lock 5
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