This short video elaborates upon the information displayed in a boxplot.

The original slides are not available.

This document is linked from Boxplots.

]]>In slide 7, there is an extra “the” in the third bullet. “If we standardize an entire **the **variable, the new variable will…”

This extremely short video contains an overview of the five-number summary.

The original slides are not available.

Transcript – Live Five-Number Summary

This document is linked from Measures of Position.

]]>**Related SAS Tutorials**

- 5A – (3:01) Numeric Measures using PROC MEANS

**Related SPSS Tutorials**

- 5A – (8:00) Numeric Measures using EXPLORE

Although not a required aspect of describing distributions of one quantitative variable, we are often interested in where a particular value falls in the distribution. Is the value unusually low or high or about what we would expect?

Answers to these questions rely on measures of position (or location). These measures give information about the distribution but also give information about how individual values relate to the overall distribution.

A common measure of position is the percentile. Although there are some mathematical considerations involved with calculating percentiles which we will not discuss, you should have a basic understanding of their interpretation.

In general the *P*-th percentile can be interpreted as a location in the data for which approximately *P*% of the other values in the distribution fall below the *P*-th percentile and (100 –*P*)% fall above the *P*-th percentile.

The quartiles Q1 and Q3 are special cases of percentiles and thus are measures of position.

The combination of the five numbers (min, Q1, M, Q3, Max) is called the **five number summary**, and provides a quick numerical description of both the center and spread of a distribution.

Each of the values represents a measure of position in the dataset.

The min and max providing the boundaries and the quartiles and median providing information about the 25th, 50th, and 75th percentiles.

Standardized scores, also called z-scores use the mean and standard deviation as the primary measures of center and spread and are therefore most useful when the mean and standard deviation are appropriate, i.e. when the distribution is reasonably symmetric with no extreme outliers.

For any individual, the **z-score** tells us how many standard deviations the raw score for that individual deviates from the mean and in what direction. A positive z-score indicates the individual is above average and a negative z-score indicates the individual is below average.

To calculate a z-score, we take the individual value and subtract the mean and then divide this difference by the standard deviation.

Measures of position also allow us to compare values from different distributions. For example, we can present the percentiles or z-scores of an individual’s height and weight. These two measures together would provide a better picture of how the individual fits in the overall population than either would alone.

Although measures of position are not stressed in this course as much as measures of center and spread, we have seen and will see many measures of position used in various aspects of examining the distribution of one variable and it is good to recognize them as measures of position when they appear.

]]>

Use the boxplot below to answer the following questions.

This document is linked from Boxplots.

]]>(Link to Best Actor Oscar Data).

Recall the results from the previous activity:

The boxplot below displays the ages of Best Actor Oscar winners (1970-2001).

Label the different numerical measures as they are depicted in the boxplot.

This document is linked from Boxplots.

]]>(Link to Best Actor Oscar Data).

The results below were obtained using SPSS. Use the output to answer the following questions

This document is linked from Boxplots.

]]>**Related SAS Tutorials**

- 5B – (4:05) Creating Histograms and Boxplots using SGPLOT

**Related SPSS Tutorials**

- 5B – (2:29) Creating Histograms and Boxplots

Now we introduce another graphical display of the distribution of a quantitative variable, the **boxplot**.

So far, in our discussion about measures of spread, some key players were:

- the extremes (min and Max), which provide the range covered by all the data; and
- the quartiles (Q1, M and Q3), which together provide the IQR, the range covered by the middle 50% of the data.

Recall that the combination of all five numbers (min, Q1, M, Q3, Max) is called the **five number summary**, and provides a quick numerical description of both the center and spread of a distribution.

We will continue with the Best Actress Oscar winners example (Link to the Best Actress Oscar Winners data).

34 34 26 37 42 41 35 31 41 33 30 74 33 49 38 61 21 41 26 80 43 29 33 35 45 49 39 34 26 25 35 33

The five number summary of the age of Best Actress Oscar winners (1970-2001) is:

min = 21, Q1 = 32, M = 35, Q3 = 41.5, Max = 80

To sketch the boxplot we will need to know the 5-number summary as well as identify any outliers. We will also need to locate the largest and smallest values which are not outliers. The stemplot below might be helpful as it displays the data in order.

Now that you understand what each of the five numbers means, you can appreciate how much information about the distribution is packed into the five-number summary. All this information can also be represented visually by using the boxplot.

The boxplot graphically represents the distribution of a quantitative variable by visually displaying the five-number summary and any observation that was classified as a suspected outlier using the 1.5(IQR) criterion.

(Link to the Best Actress Oscar Winners data).

- The central box spans from Q1 to Q3. In our example, the box spans from 32 to 41.5. Note that the width of the box has no meaning.

- A line in the box marks the median M, which in our case is 35.

- Lines extend from the edges of the box to the smallest and largest observations that were not classified as suspected outliers (using the 1.5xIQR criterion). In our example, we have no low outliers, so the bottom line goes down to the smallest observation, which is 21. Since we have three high outliers (61,74, and 80), the top line extends only up to 49, which is the largest observation that has not been flagged as an outlier.

- outliers are marked with asterisks (*).

To summarize: the following information is visually depicted in the boxplot:

- the five number summary (blue)
- the range and IQR (red)
- outliers (green)

As we learned earlier, the distribution of a quantitative variable is best represented graphically by a histogram. Boxplots are most useful when presented side-by-side for comparing and contrasting distributions from two or more groups.

So far we have examined the age distributions of Oscar winners for males and females separately. It will be interesting to compare the age distributions of actors and actresses who won best acting Oscars. To do that we will look at side-by-side boxplots of the age distributions by gender.

Recall also that we found the five-number summary and means for both distributions. For the Best Actress dataset, we did the calculations by hand. For the Best Actor dataset, we used statistical software, and here are the results:

- Actors: min = 31, Q1 = 37.25, M = 42.5, Q3 = 50.25, Max = 76
- Actresses: min = 21, Q1 = 32, M = 35, Q3 = 41.5, Max = 80

Based on the graph and numerical measures, we can make the following comparison between the two distributions:

**Center:** The graph reveals that the age distribution of the males is higher than the females’ age distribution. This is supported by the numerical measures. The median age for females (35) is lower than for males (42.5). Actually, it should be noted that even the third quartile of the females’ distribution (41.5) is lower than the median age for males. We therefore conclude that in general, actresses win the Best Actress Oscar at a younger age than actors do.

**Spread:** Judging by the range of the data, there is much more variability in the females’ distribution (range = 59) than there is in the males’ distribution (range = 45). On the other hand, if we look at the IQR, which measures the variability only among the middle 50% of the distribution, we see more spread in the ages of males (IQR = 13) than females (IQR = 9.5). We conclude that among all the winners, the actors’ ages are more alike than the actresses’ ages. However, the middle 50% of the age distribution of actresses is more homogeneous than the actors’ age distribution.

**Outliers:** We see that we have outliers in both distributions. There is only one high outlier in the actors’ distribution (76, Henry Fonda, On Golden Pond), compared with three high outliers in the actresses’ distribution.

In order to compare the average high temperatures of Pittsburgh to those in San Francisco we will look at the following side-by-side boxplots, and supplement the graph with the descriptive statistics of each of the two distributions.

Statistic | Pittsburgh | San Francisco |
---|---|---|

min | 33.7 | 56.3 |

Q1 | 41.2 | 60.2 |

Median | 61.4 | 62.7 |

Q3 | 77.75 | 65.35 |

Max | 82.6 | 68.7 |

When looking at the graph, the similarities and differences between the two distributions are striking. Both distributions have roughly the same center (medians are 61.4 for Pitt, and 62.7 for San Francisco). However, the temperatures in Pittsburgh have a much larger variability than the temperatures in San Francisco (Range: 49 vs. 12. IQR: 36.5 vs. 5).

The practical interpretation of the results we obtained is that the weather in San Francisco is much more consistent than the weather in Pittsburgh, which varies a lot during the year. Also, because the temperatures in San Francisco vary so little during the year, knowing that the median temperature is around 63 is actually very informative. On the other hand, knowing that the median temperature in Pittsburgh is around 61 is practically useless, since temperatures vary so much during the year, and can get much warmer or much colder.

Note that this example provides more intuition about variability by interpreting small variability as consistency, and large variability as lack of consistency. Also, through this example we learned that the center of the distribution is more meaningful as a typical value for the distribution when there is little variability (or, as statisticians say, little “noise”) around it. When there is large variability, the center loses its practical meaning as a typical value.

- The five-number summary of a distribution consists of the median (M), the two quartiles (Q1, Q3) and the extremes (min, Max).

- The five-number summary provides a complete numerical description of a distribution. The median describes the center, and the extremes (which give the range) and the quartiles (which give the IQR) describe the spread.

- The boxplot graphically represents the distribution of a quantitative variable by visually displaying the five number summary and any observation that was classified as a suspected outlier using the 1.5(IQR) criterion. (Some software packages indicate extreme outliers with a different symbol)

- Boxplots are most useful when presented side-by-side to compare and contrast distributions from two or more groups.