1. Describing Data: One Variable
STAT 101
Dr. Kari Lock Morgan
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)

2. Announcements Homework 1 due now – turn it in according to lab section
Clicker grading starts today!

3. Why not always randomize?
Randomized experiments are ideal, but sometimes not ethical or possible
Often, you have to do the best you can with data from observational studies
Example: research for the Supreme Court case as to whether preferences for minorities in university admissions helps or hurts the minority students

4. Randomization in Data Collection
Was the explanatory variable randomly assigned?
Was the sample randomly selected?
Yes No
Yes No
Possible to generalize to the population
Should not generalize to the population
Possible to make conclusions about causality
Can not make conclusions about causality

5. Two Fundamental Questions in Data Collection
Random sample???
Population
Sample
Randomized experiment???
DATA

6. Randomization
Doing a randomized experiment on a random sample is ideal, but rarely achievable
If the focus of the study is using a sample to estimate a statistic for the entire population, you need a random sample, but do not need a randomized experiment (example: election polling)
If the focus of the study is establishing causality from one variable to another, you need a randomized experiment and can settle for a non- random sample (example: drug testing)

7. Review from Last Class Association does not imply causation!
In observational studies, confounding variables almost always exist, so causation cannot be established
Randomized experiments involve randomly determining the level of the explanatory variable
Randomized experiments prevent confounding variables, so causality can be inferred
A control or comparison group is necessary
The placebo effect exists, so a placebo and blinding should be used

8. Descriptive Statistics
The Big Picture
Population
Sampling
Sample
Statistical Inference
Descriptive Statistics

9. 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)

10. 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?

11. Cell Phones Which type of cell phone do you own? iPhone Blackberry
Android
iPhone
Blackberry
Non-smartphone
No cell phone

12. Frequency Table
US data: 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
R: table(x)

13. The proportion in a category is found by
Proportion for a sample: 𝑝 (“p-hat”)
Proportion for a population: p

14. Proportion What proportion of adults sampled do not own a cell phone?
𝑝 = =0.13
Android
458
iPhone
437
Blackberry
141
Non Smartphone
924
No cell phone
293
Total
2253
or 13%
Proportions and percentages can be used interchangeably

15. Relative Frequency Table
A relative frequency table shows the proportion of cases that fall in each category
All the numbers in a relative frequency table sum to 1
Android
0.203
iPhone
0.194
Blackberry
0.063
Non Smartphone
0.410
No cell phone
0.130
R: table(x)/length(x)

16. Bar Chart/Plot/Graph
In a bar chart, the height of the bar is the number of cases falling in each category
R: barchart(x)

17. 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))

18. StatKey

19. Summary: One Categorical Variable
Summary Statistics
Proportion
Frequency table
Relative frequency table
Visualization
Bar chart
Pie chart

20. One Quantitative Variable
World gross for all Hollywood movies
HollywoodMovies2011
More graphics on profits for Hollywood movies

21. HollywoodMovies2011

22. Dotplot
In a dotplot, each case is represented by a dot and dots are stacked.
Easy way to see each case
Highest is Harry Potter and the Deathly Hallows Part 2
Second is Transformers
Third: Pirates of the Caribbean: On Stranger Tides
units: Millions of $

23. Histogram
The height of the each bar corresponds to the number of cases within that range of the variable
R: hist(x)

24. Histogram vs Bar Chart This is a Histogram Bar chart Other
I have no idea

25. Histogram vs Bar Chart This is a Histogram Bar chart Other
I have no idea

26. 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

27. Shape
Long right tail
Symmetric
Right-Skewed
Left-Skewed

28. 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 x1 , x2 , …, xn represent the n values of the variable x
x1 = , x2 = , x3 = , …

29. Mean
The mean or average of the data values is 𝑚𝑒𝑎𝑛= 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠
𝑚𝑒𝑎𝑛= 𝑥 1 + 𝑥 2 +…+ 𝑥 𝑛 𝑛 = 𝑥 𝑛
Sample mean: 𝑥
Population mean:  (“mu”)
R: mean(x)

30. 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)

31. Measures of Center
m = 76.66
Mean is “pulled” in the direction of skewness
 =150.74
World Gross (in millions)

32. Skewness and Center
A distribution is left-skewed. Which measure of center would you expect to be higher?
Mean
Median
The mean will be pulled down towards the skewness (towards the long tail).

33. Outlier
An outlier is an observed value that is notably distinct from the other values in a dataset.

34. World Gross (in millions)
Outliers
Transformers
Harry Potter
Pirates of the Caribbean
World Gross (in millions)

35. 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

36. 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

37. Standard Deviation
The standard deviation for a quantitative variable measures the spread of the data
𝑠= 𝑥− 𝑥 2 𝑛−1
Sample standard deviation: s
Population standard deviation:  (“sigma”)
R: sd(x)

38. 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

39. Standard Deviation
Both of these distributions are bell-shaped

40. 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

41. The 95% Rule

42. The 95% Rule
The normal distribution app on statkey is a good way to demonstrate the 95% rule – ask students to pick any mean and sd, and then have them guess what the bounds for the middle 95% (can get by clicking on two tail) will be
StatKey

43. The 95% Rule
The standard deviation for hours of sleep per night is closest to 1 2 4
I have no idea

44. The z-score for a data value, x, is 𝑧= 𝑥− 𝑥 𝑠
For a population, 𝑥 is replaced with µ and s is replaced with 
Values farther from 0 are more extreme

45. 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

46. 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
ACT score of 28
SAT score of 2100
I don’t know

47. Other Measures of Location
Maximum = largest data value
Minimum = smallest data value
Quartiles:
Q1 = median of the values below m.
Q3 = median of the values above m.

48. Five Number Summary Five Number Summary: Min Max Q1 Q3 m 25%
R: summary(x)

49. Five Number Summary
> summary(study_hours)
Min. 1st Qu. Median 3rd Qu. Max.
The distribution of number of hours spent studying each week is
Symmetric
Right-skewed
Left-skewed
Impossible to tell

50. 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

51. Five Number Summary Five Number Summary: Min Max Q1 Q3 m 25%
0th percentile
25th percentile
50th percentile
75th percentile
100th percentile

52. Measures of Spread Range = Max – Min
Interquartile Range (IQR) = Q3 – Q1
Is the range resistant to outliers?
Yes No
Is the IQR resistant to outliers?
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.

53. Comparing Statistics Measures of Center: Measures of Spread:
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

54. 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 Q1 – 1.5(IQR) or
Larger than Q (IQR)

55. Boxplot
Outliers
Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier Q3
Median Q1
R: boxplot(x)

56. Boxplot
Which boxplot goes with the histogram of waiting times for the bus?
(a)
(b)
(c)
The data do not show any low outliers.

57. StatKey

58. 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

59. To Do
Read Sections 2.1, 2.2, 2.3, 2.4
Do HW 2 (due Wednesday, 1/29)