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Statistics: Unlocking the Power of Data Lock^5 STAT 101 Dr. Kari Lock Morgan 9/6/
Describing Data:
One Variable
SECTIONS 2.1, 2.2, 2.3, 2.
- 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
Sample Sampling
Statistical
Inference Descriptive Statistics
Statistics: Unlocking the Power of Data 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? iPhone? Blackberry? Non- smartphone? No cell phone? Statistics: Unlocking the Power of Data Lock^5
Frequency Table
R: table(x)
- A frequency table shows the number of cases that fall in each category: Android 458 iPhone 437 Blackberry 141 Non Smartphone 924 No cell phone 293 Total 2253 Statistics: Unlocking the Power of Data Lock^5
Proportion
The proportion in a category is found by 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 =
Proportion for a sample: 𝑝 (“p-hat”) Proportion for a population: p
Statistics: Unlocking the Power of Data Lock^5
Proportion
What proportion of adults sampled do not own a cell phone? Android 458 iPhone 437 Blackberry 141 Non Smartphone 924 No cell phone 293 Total 2253
or 13% Proportions and percentages can be used interchangeably Statistics: Unlocking the Power of Data Lock^5
Relative Frequency Table
A relative frequency table shows the proportion of cases that fall in each category R: table(x)/length(x) Android 0. iPhone 0. Blackberry 0. Non Smartphone 0. No cell phone 0. All the numbers in a relative frequency table sum to 1 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
StatKey
www.lock5stat.com/statkey Statistics: Unlocking the Power of Data Lock^5
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
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
Symmetric Right-Skewed Left-Skewed Long right tail Statistics: Unlocking the Power of Data 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
The mean or average of the data values is 𝑚𝑒𝑎𝑛 = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 Sample mean: 𝑥 Population mean: (“mu”)
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 are 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 m = 76. =150. Mean is “pulled” in the direction of skewness Measures of Center 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 The mean will be pulled down towards the skewness (towards the long tail). Statistics: Unlocking the Power of Data 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 World Gross (in millions) Harry Pirates of the^ Transformers Potter Caribbean 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, Without Harry Potter $141,889,900 $75,009, 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
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? z-scores beyond - 2 or 2 can be considered extreme
- 2 and 2 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) ACT score of 28 b) SAT score of 2100 c) I don’t know (^28 21 7) 1. z (^) 5 5 (^2100 1500 600) 1. z (^) 325 325 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 m Q 3 Max 25% 25% 25% 25% R: summary(x) Statistics: Unlocking the Power of Data Lock^5
Five Number Summary
The distribution of number of hours spent studying each week is a) Symmetric b) Right-skewed c) Left-skewed d) Impossible to tell
summary(study_hours) Min. 1st Qu. Median 3rd Qu. Max. 2.00 10.00 15.00 20.00 69. 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: 91st percentile SAT score of 2100: 97th percentile
Statistics: Unlocking the Power of Data Lock^5
Five Number Summary
Five Number Summary: Min Q 1 m Q 3 Max 25% 25% 25% 25% 0 th percentile 100 th percentile 50 th percentile 75 th percentile 25 th percentile Statistics: Unlocking the Power of Data Lock^5
Measures of Spread
Range = Max – Min
Interquartile Range (IQR) = Q 3 – Q 1
Is the range resistant to outliers? a) Yes b) No Is the IQR resistant to outliers? a) Yes b) No The range depends entirely on the most extreme values. The IQR is based off the middle 50% of the data, which will not 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
Q^ Median 1 Q 3 middle 50% of data
- Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier Outliers R: boxplot(x) Statistics: Unlocking the Power of Data Lock^5
Boxplot
Which boxplot goes with the histogram of waiting times for the bus? Histogram of Bus Bus Frequency 0 5 10 15 20 0 10 20 (a) (b) (c) The data do not show any low outliers.
