## A Statistical Look at the Amendments to the United States Constitution

The Constitution of the United States is the supreme law of the United States. The framers of the Constitution were aware that changes to the Constitution would be necessary from time to time. A two-part process was set up for proposing and ratifying amendments to the Constitution. There are currently 27 amendments to the Constitution. The first 10 amendments are known collectively as the Bill of Right. The most recent amendment, the 27th, was ratified in 1992. On average, how long did it take to ratify an amendment (the average time from proposal to enactment)? How speedy or lengthy was the process? In this post, we first informally discuss the time from proposal to enactment for these amendments. Then we look at the 27 amendments in more details through descriptive statistics, which entails describing the data graphically and using numerical summaries. We also discuss the notion of resistant statistic. In particular, median is resistant to extreme data values while mean is not.

The following tables show the dates for the amendments and the time span (in months) from proposal to enactment.

Data Tables
$\displaystyle \text{Table 1} \begin{pmatrix} \text{Amendment}&\text{Proposal Date}&\text{ }&\text{Enactment Date}&\text{Number of Months} \\\text{ }&\text{ }&\text{ }&\text{ } \\\text{1st}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{2nd}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{3rd}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{4th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{5th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{6th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{7th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{8th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{9th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{10th}&\text{September 25, 1789}&\text{ }&\text{December 15, 1791}&27 \\\text{11th}&\text{March 4, 1794}&\text{ }&\text{February 7, 1795}&11 \\\text{12th}&\text{December 9, 1803}&\text{ }&\text{June 15, 1804}&6 \\\text{13th}&\text{January 31, 1865}&\text{ }&\text{December 6, 1865}&10 \\\text{14th}&\text{June 13, 1866}&\text{ }&\text{July 9, 1868}&25 \\\text{15th}&\text{February 26, 1869}&\text{ }&\text{February 3, 1870}&11 \end{pmatrix}$

$\displaystyle \text{Table 2} \begin{pmatrix} \text{Amendment}&\text{Proposal Date}&\text{ }&\text{Enactment Date}&\text{Number of Months} \\\text{ }&\text{ }&\text{ }&\text{ } \\\text{16th}&\text{July 12, 1909}&\text{ }&\text{February 3, 1913}&43 \\\text{17th}&\text{May 13, 1912}&\text{ }&\text{April 8, 1913}&11 \\\text{18th}&\text{December 18, 1917}&\text{ }&\text{January 16, 1919}&13 \\\text{19th}&\text{June 4, 1919}&\text{ }&\text{August 18, 1920}&14 \\\text{20th}&\text{March 2, 1932}&\text{ }&\text{January 23, 1933}&11 \\\text{21st}&\text{February 20, 1933}&\text{ }&\text{December 5, 1933}&10 \\\text{22nd}&\text{March 24, 1947}&\text{ }&\text{February 27, 1951}&47 \\\text{23rd}&\text{June 16, 1960}&\text{ }&\text{March 29, 1961}&9 \\\text{24th}&\text{September 14, 1962}&\text{ }&\text{January 23, 1964}&16 \\\text{25th}&\text{July 6, 1965}&\text{ }&\text{February 10, 1967}&19 \\\text{26th}&\text{March 23, 1971}&\text{ }&\text{July 1, 1971}&3 \\\text{27th}&\text{September 25, 1789}&\text{ }&\text{May 7, 1992}&2432 \end{pmatrix}$

An Informal Look
The time it took an amendment to become law varied. The 26th amendment (establishing 18 as the national voting age) only took 3 months and 8 days. No doubt, this one was propelled by popular demand. The longest one was the 27th amendment (restricting the power of Congress to set its own salary) took over 202 years! Obviously the 27th amendment is an outlier. The next longest one was the 22nd amendment (presidential term limit) and it took 3 years and 11 months. In fact, nine of the amendments took a year or less to become law. Except for 16th, 22nd and 27th, all of the amendments took less than 3 years to become the law of the land. As long as an amendment made it out of the starting gate, the process to enactment was quite speedy.

Descriptive Statistics
Since the 27th amendment is an extreme data value, we exclude it from the analysis. It seems that the road to enactment for the 27th amendment had a long and tortuous journey, which we do not want to focus on here. Including the 27th will make any graph excessively wide, thus providing no insight. So we will focus on the overall pattern of the first 26 amendments.

We use this strategy for data analysis: 1. start with a graphical display of the data, 2. look for the overall pattern and note any deviations from that pattern, and 3, use numerical summaries to describe certain aspects of the data (see [1]). All the numerical summaries are calculated using the calculator TI83 plus.

To display one-variable data such as the time to enactment for the amendments to the Constitution, we can use histogram, stemplot, or boxplot. In this post, we use histogram and stemplot to display the data. Once we have a graph, we describe the distribution of the time to enactment (the overall pattern) by noting its shape, center and spread and noting any outliers (individual values that fall outside of the overall pattern). Then we discuss the numerical summaries of center and spread in more details.

The following is a frequency distribution of the time to enactment data. Figure 1 is a histogram and Figure 2 is a stemplot.

$\displaystyle \begin{pmatrix} \text{Time (months)}&\text{Frequency} \\\text{ }&\text{ } \\\text{0 to 9.9}&3 \\\text{10 to 19.9}&10 \\\text{20 to 20.9}&11 \\\text{30 to 30.9}&0 \\\text{40 to 49.9}&2 \\\text{ }&\text{ }& \\\text{Total}&26 \end{pmatrix}$

What is the overall pattern of the distribution of the time to enactment of the 26 amendments? Shape: The distribution is roughly symmetric (it is a little skewed but the skewness is not pronounced). Center: The midpoint of the distribution is 22 (taking the average of the 13th and 14th leaves in the stemplot). Spread: The spread is from 3 to 47 months. There are no obvious outliers or other striking deviations from the overall pattern.

What are some numerical summaries that we can use to describe the distribution of the time to enactment for the amendments? There are two types of measures to consider, the measures of center and the measures of spread. A measure of center is a numerical summary that attempts to describe what a typical data value might look like. A measure of spread is a numerical summary that describes the degree to which the data are spread out.

Two common measures of center are mean and median. The mean time to enactment for an amendment is $\overline{x}=20.35$ months. The median time to enactment is 22 months. With either notion of center, the average time to enactment is just a little under two years.

A measure of center alone is an incomplete description of the distribution. Two distributions can be very different even when they have the same median (e.g. one is a skewed distribution and the other has the shape of a bell curve). So we should include a measure of spread in describing a distribution. One numerical summary that contain both a measure of center and a measure of spread is the 5-number summary.

The 5-number summary for distribution of the time to enactment is 3, 11, 22, 27, 47 (minimum, first quartile $Q_1$, median, third quartile $Q_3$, maximum). The measure of center and the measure of spread contained in the 5-number summary are the median and the interquartile range (defined as $IQR=Q_3-Q_1$). The IQR for the time to enactment distribution is 16 months. Thus the middle 50% of the time to enactment was between 11 months and 27 months.

Another measure of spread is the standard deviation ($s=10.96$ months for the time to enactment data). The standard deviation measures spread by calculating how far the data points are from the mean.

Resistant Numerical Summaries
A numerical summary is resistant if it is not sensitive to the influence of a few extreme data values. In other words, a few extreme data values do not affect the value of the numerical summary significantly. Both the median and IQR are resistant, while the mean and the standard deviation are not. To see this, we can add the 27th amendment back in.

With the 27th amendment in the data set, the 5-number summary is 3, 11, 25, 27, 2432. Thus the median only changes slightly (25 vs. 22) and IQR is unchanged. However, the mean is now $\overline{x}=109.67$ months (vs. 20.35 before) and the standard deviation is $s=464.25$ (vs. 10.96 before).

For any reasonably symmetric distribution that has no outliers, mean and standard deviation are better measures of center and spread, respectively. For skewed distributions, we should not use mean and standard deviations and should use measures that are resistant numerical summaries such as median and IQR (or 5-number summary). For a skewed distribution, the median is a more reliable indication of what a typical data value look like.

Since the time to enactment distribution is roughly symmetric (without the 27th amendment), the combination of mean and standard deviation is an appropriate description of the distribution.

Reference

1. Moore. D. S., McCabe G. P., Craig B. A., Introduction to the Practice of Statistics, 6th ed., W. H. Freeman and Company, New York, 2009