Using histograms for visualizing statistics from playing a hundred games of solitaire

 

 

 

 

 

 

 

 

 

 

 

 

 

It often is helpful to show the details of data rather than just a mean and standard deviation. Recently I played a hundred games of Solitaire to generate the game times shown above via a histogram (created using Excel with bins ten seconds wide). The mean time to complete a game is 221 seconds (3 minutes and 41 seconds), and the standard deviation is 31 seconds. But the range goes from 160 to 335 seconds. The distribution is roughly symmetrical, with three longer outliers.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I also kept track of the number of moves per game, as shown above via another histogram using a bin width of five. The mean is 133 moves, and the standard deviation is 13 moves. The range goes from 116 to 170 moves. The distribution is skewed positively (to the right) with two outliers.

 

Two very useful articles in the August 2025 issue of Toastmaster magazine on Five Tips for Depicting Data and Making Your Data Presentations Come to Life

 

 

 

 

 

 

 

 

 

The August 2025 issue of Toastmaster magazine has two very useful articles. One by Florian Bay on pages 20 to 23 is titled 5 Tips for Depicting Data and subtitled How to use numbers and graphs to create a compelling story. His five headings are:

 

Be Clear on Your Purpose

Select the Right Visual

Remove the Clutter

Engage with Colors but Don’t Create Rainbows

Use Text to Guide Your Viewers

 

The first two line and column bar charts, at the bottom of Page 20 are missing a label for the vertical axis, which should be Number of UNESCO Sites. As shown above, I have added it in red.

 

Regarding pie charts, he says:

 

“Visualizing proportions? Then use a pie chart, but only if you’ve got two or three slices to share; any more than that and the proportions get hard to interpret. Stacked bar charts can also be useful.”

 

I don’t think pie charts ever are useful, as I discussed in a blog post on September 16, 2008 titled Pie charts do not speak clearly; they just mumble and one on July 29, 2022 titled Is there any excuse for using a pie chart?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

And his example with Yearly Passengers in London Heathrow Airport 2005-2024 (shown above) has the numbers of years shown vertically rather than horizontally. My Excel version omits the bullseyes around the pandemic data and just shows numbers for all those data points. (On July 4, 2017 I blogged about Why is your audience tilting their heads sideways?)

 

The second article by Charlene Phua following on pages 24 and 25 is titled Make Your Data Presentations Come to Life and subtitled Avoid these 4 missteps to ensure your audience stays engaged. Her four headings are:

 

Including data that isn’t relevant to topic or audience

Having an inaccurate gauge of your audience’s understanding

Providing data with no context or comparison

Neglecting other aspects of your performance

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For the third she uses the example of annual global carbon dioxide emissions. She didn’t mention a number, but in 2022 they are 38,521,997,860 tons. We can ask What’s in it for me? The global Population is  8,021,407,192 people so it is 4.802 tons per capita, or 9605 pounds. Per day it is 26.3 pounds, and converting that to a cube of dry ice, it is one 7.75 inches on a side, as shown above.

 

On December 23, 2024 I blogged about How should we present a huge number like the two billion dollars earned by the Taylor Swift Eras tour? And on July 12, 2016 I blogged about How to make statistics understandable. Also, on August 17, 2011 I posted on How to make a large number incomprehensible or comprehensible and on July 15, 2011 I blogged on What can we say about a really big hole in the ground?

 

 

A visit from the mythical Bar Chart Police could have improved the 2025 book Speak, Memorably: the art of captivating an audience

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I have been reading a 2025 book by Bill McGowan and Juliana Silva which is titled Speak, Memorably: the art of captivating an audience. It’s not awful, but it’s not great either. There is no index. On August 2, 2025 I posted about A bogus car story from the 2025 book Speak, Memorably.

 

On page 10 of that book they state that:

 

“….in the chapters that follow, we commissioned a survey by the Clarity Media Group of over one hundred people. The respondent pool was comprised of full and part-time employees between the ages of twenty-five and sixty-five.”

 

There are 31 horizontal bar charts shown in this book on pages 11, 15,21, 27, 36, 38, 55, 58,60, 71, 113, 114, 135, 139, 153, 155, 168, 183, two on 185, 194, 197, 199, 204, 214, 215, 226, 228, 229, 243, and 244. They are shown in monochrome, and only are 3.3” wide by 1.5” high. The percent is labeled at the center of the bar, and the description of the question is to the right of the bar.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

My slightly different Excel version of the chart on page 11 for Do You Consider Your Public Speaking Skills at Work Important? is shown above. But those bars are not presented in what should be their order, with the most significant and largest percentage placed at the top. My revised version also is shown above.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Another example from page 168 about How Do You Determine What Your Audience Wants to Know? and a corrected version are shown above.

 

18 of 31 charts have this incorrect order, those on pages 11, 21, 27, 36, 113, 139, 153, 155, 168, 183, 185 bottom, 194, 199, 204, 214, 215, 228, and 229.

 

 

 

 

 

 

 

 

 

 

 

 

And as shown above, the chart on page 21 for How Often Do You Know with Certainty the First and Last Sentence of Each Slide? has silly labels from top to bottom of: Never, Rarely, Sometimes, and Frequently. That is opposite to what should be: Frequently, Sometimes Rarely, Never. They used the same sequence on pages 27, 113, 139, 155, 199, and 215. Also the chart on page 194 has Never, Seldom, Occasionally, Frequently.

 

Back on September 6, 2022 I blogged that we should Beware of surveys with small sample sizes, which have large margins of error. There is a Wikipedia page about Margin of error. For a typical national survey with a sample size of 1,000 the margin of error is 3.1%. For a sample size of 100 the margin of error instead  is 9.8% (as is shown above in my revised version for page 11).

 

Is the difference between the largest percentage and the second largest percentage twice the margin of error (or 20%)? That only is true for ten charts, those on pages 36, 55, 58, 71, 135, 139, 168, 183, 185 top, and 215. Not the other 21 charts!

 

I first mentioned the mythical Bar Chart Police (and showed a version of their badge) in a post on November 3, 2023 titled Five of six British age groups are more confident about public speaking after COVID.

 

Seeing data using exploratory data analysis and histograms

 

 

 

 

 

 

 

 

 

 

 

When I get bored, I play solitare – the online version from TheSolitaire.com. Recently I got curious about how much time and how many moves it was taking me to complete a game. What would the averages be? Would their shape just follow a Normal Distribution - the symmetrical bell curve shown above. It has roughly two thirds of the data (68.3%) in a range plus or minus one standard deviation from the mean.  

 

 

 

 

 

 

 

 

Or would there instead be a skew distribution, as shown above? I did some exploratory data analysis. I kept track of a hundred games, and then used Microsoft Excel to plot histograms.

 

 

 

 

 

 

 

 

 

 

 

A histogram for time to complete is shown above. The mean time to complete was 208 seconds (3 minutes and 28 seconds) and the standard deviation was 27.6 seconds.  Columns in this histogram are twenty seconds wide. There are roughly two and a half to the left of the mean, but four and a half to the right.

 

 

 

 

 

 

 

 

 

 

 

 

Another histogram, for the number of moves to complete, is shown above. The mean number of moves was 134 and the standard deviation was 12 moves. Columns in this histogram are ten moves wide. There are three to the left of the mean, but four to the right.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Both histograms have a positive skew. They are shaped like the slide on a children’s playground, as shown above via a cartoon.

 

Another misleading article about government spending from the Idaho Freedom Foundation

 

 

 

 

 

 

 

At the Idaho Freedom Foundation on December 31, 2024 there is a misleading article by Fred Birnbaum titled Want limited government? Control the spending! He preaches that:

 

“Legislative leadership needs to set both state and federal (all funds) spending limits so that overall spending does not grow at all in the coming year, meaning a 0% increase in all funds appropriations from Fiscal Year 2025 to Fiscal Year 2026. Is this reasonable? Absolutely!”

 

 

 

 

 

 

 

Fred has a pair of tables showing how state spending grew from fiscal 2020 to 2025, and how federal spending changed from fiscal 2019 to 2024. He shows a totally unnecessary seven or eight significant figures. Graphs would have been more effective for showing changes. Fred claims that there have been large increases:

 

“What we see is that state spending, excluding federal dollars, has increased 53% over five years at a compound annual growth rate (CAGR) of 8.8%. Now let’s look at the federal dollars (all federal spending, not just federal dollars for Idaho) for the last five years – note that a one-year offset is used because of differences in fiscal years and the fact that COVID spending first washed through the federal government before hitting the states.

What we see is that federal spending is up 56%, and the CAGR is 9.3%. This means that Idaho is growing spending virtually as fast as the profligate federal government. Who would have thought this?”

 

Fred summarized those tables with a five-year % increase and a CAGR. But what was going on from year to year?

 

 

 

 

 

 

 

 

 

 

 

The graph shown above made using Excel displays the state spending data. Year to year % increases in spending were 4.75%, 4.64%, 18.6%, 11.5%, and 5.35%. The first two were around 5%, but the next two were much larger. The five-year % increase was 53% and the CAGR was 8.8%.

  

 

 

 

 

 

 

 

 

 

 

 

 

 

Another graph shows the federal data he tabulated in red. Year to year % changes in spending were 47.4%, 4.1%, -8.0%, -2.2% and 13.1%. The five-year % increase was 56% and the CAGR was 9.3%. With a huge range of -2.2% to 47.4% the CAGR tells us very little. But Fred did not link to a source for that federal data, so I went looking for one. I found a United States Treasury Department FiscalData web page titled How much has the U.S. government spent this year? which has a graphic titled Government Spending and the U. S. Economy (GDP, FY 2015 -2024 Inflation Adjusted – 2024 dollars. I also have plotted that federal data in blue. It does NOT agree with what Fred tabulated. The five-year % increase only was 23.6% and the CAGR just was 3.9%. But the maximum change, from 2019 to 2020, was by 45.4%. Using the five-year % increase gives a much more modest change of 23.6%, where Idaho spending increased over twice as much as federal spending did!

 

 

 

 

 

 

 

 

 

 

 

 

That Treasury web page has more spending data, back to 2015. All those results are shown above in yet another graph. We can look at 2020, 2021, and 2022 as being a COVID-19 hump. Then we can draw a line with a 5% growth rate through 2018, 2019, 2023 and 2024. A 5% growth rate would compensate for inflation and be much more reasonable than Fred’s zero.   

 

The cartoon with money bags was adapted from this one at Openclipart.

 

Psychotherapist Jonathan Berent fumbles some statistics about social anxiety and fear of public speaking

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Jonathan Berent has been a psychotherapist since 1978 and wrote two self-help books. He wrote an article at KevinMd on November 2, 2024 titled When high achievers and isolated adults share a common enemy: social anxiety containing the following claims:

 

“The Anxiety and Depression Association of America estimates that 15 million adults have social anxiety. This number swells when you consider stats, such as the Chapman University Survey of American Fears (2019), which found that 10.7 percent of Americans have a public speaking phobia, and 21.2 percent to 23.3 percent fear public speaking. Similarly, a 2001 Gallup poll found that 40 percent of adults in the United States fear public speaking.”

 

The US Census in 2020 found there were 258.3 million adults, so the 15 million estimated by the Anxiety & Depression Association of America is just 5.8%. That’s lower than what the National Institute of Mental Health web page on Social Anxiety Disorder says: 7.1% in the last year or 12.7% in a lifetime.

 

 

 

 

 

 

 

 

 

 

 

The Chapman Survey of American Fears reported percent fears at four levels: Very Afraid, Afraid, Slightly Afraid, and Not Afraid. Two bar charts with all their data for public speaking from 2014 to 2024 made using Excel are shown above. In 2019 12.8% were Very Afraid, and 10.7% never appeared anywhere. And in 2019 31.2% were Very Afraid or Afraid, not 21.2%. In 2017 23.3% were Very Afraid or Afraid. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Chapman Survey of American Fears did not measure phobia, which is different from Very Afraid. Writing in 2024 why did Jonathan refer to percentages from back in 2019?

 

And the article by Geoffrey Brewer at Gallup on March 19, 2001 is titled Snakes Top List of Americans’ Fears - 51% feared snakes versus 40% for public speaking. But he also reported that in 1998 56% feared snakes while only 45% feared public speaking.

 

The cartoon was adapted from one at Wikimedia Commons.

 

Overblown claims about fears from investigators for the 2024 Chapman Survey of American Fears

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

On October 24, 2024 I blogged about how In the tenth Chapman Survey of American Fears for 2024, public speaking only was ranked #59 of 85 fears at 29.0% There is a pdf article titled Chapman Survey of American Fears 2024 – Key Findings that contains the following statement:

 

“ ‘More interesting, really, than what is in the top 10 list, is how the overall level of fear changes. For example, in the 2015 survey, only the top one, Corrupt Government Officials, had more than 50% reported afraid or very afraid. By the time you get to 2018, all 10 are over 50%. Americans are more afraid of everything,’ said Ed Day, Associate Professor of Sociology.”

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

The above statement refers to percent and thus to more Americans being afraid, NOT Americans being more afraid. But, as shown above using graphs made with Excel in 2018 there were 12 fears ranked at 50% or above. In 2019 there were 17. The number of fears rises sharply from 2016 to 2019. Then for 2020/2021 there only were 9, in 2022 there were 11, in 2023 there were only 7, and in 2024 there were 15. Similar trends (also shown above) appear when we look either at 40% or above, and 30% or above.  

 

Their discussion of Corruption says:

 

“Since the inception of the Chapman Survey of American Fears, Fear of Corrupt Government Officials (hereafter, FOC) has been the leading fear Americans reported. That is a striking fact because the surveys have spanned a decade and three presidential administrations. In the tenth wave of the survey conducted in 2024, 65.2% of Americans said that they were afraid or very afraid of corrupt government officials up more than 5% since 2023. FOC peaked at 79.6% in 2020/2021 and then declined to current levels.” 

 

 

 

 

 

 

 

 

 

 

 

 

 

See their graphic, to which I have added about the Trump administration.

 

Another article by Andre Mouchard (of the Orange County Register) published at MedicalExpress on October 28, 2024 is incorrectly titled Survey finds Americans more afraid today than at any time in history. It contains the following quotes:

 

“ ‘Fear is taking a larger and larger role in American life,’ said Chistopher Bader, a sociology professor at Chapman who has been involved in ‘American Fears’ since the beginning.

 

‘They’re afraid of more things than they used to be,’ he said.

 

‘And they’re more afraid of those things than they used to be.’….

 

 

 

 

 

 

 

 

 

 

 

 

 

 

But the overall number of fears (shown above) has ranged from 79 on 2016 to 97 in 2023. That number of questions asked is a choice made by investigators which varies, as does the sample size from about 1000 to 1500 (also shown above).  

 

Also:

 

“ ‘Stranger danger is growing,’ Bader said, referring to data that shows Americans – once viewed as optimistic and welcoming – are increasingly afraid of people they don’t know.”

 

 

 

 

 

 

 

 

 

 

 

 

But strangers were ranked between 73rd and 91st in the surveys, as shown above. Further, the percent very afraid or afraid of strangers only ranged from 7% in 2018 to 16.7% in 2024.

 

What are Americans really more afraid of? To evaluate that we need to look at Fear Scores on a scale from one to four, where 1 = Not Afraid, 2 = Slightly Afraid, 3 = Afraid, and 4 = Very Afraid. I blogged about this back on October 30, 2015 in a post titled According to the 2015 Chapman Survey of American Fears, adults are slightly less than Afraid of federal government corruption and only Slightly Afraid of Public Speaking.

 

 

 

 

 

 

 

 

 

 

 

 

 

As shown above, the Fear Score for Corrupt Government Officials ranged from 2.611 in 2015 to 3.175 (more than Afraid) in 2020/21. The Fear Score for Public Speaking ranged from 1.909 in 2017 to 2.172 in 2022 (more than Slightly Afraid). And the Fear Score for Strangers only ranged from 1.488 in 2018 to 1.750 in 2024.

 

In 2024 membership in Toastmasters International finally began to grow again, after having dropped for three years in a row

 

 

 

 

 

 

 

I have been reading the October 2024 centennial issue of Toastmaster magazine, which has an article by Paul Sterman on pages 10 to 15 titled A Century of Toastmasters International. Reaching a centennial is very impressive for an organization. On page 26 there is another article titled Toastmasters Today that begins by stating:

 

“The numbers in this statistical analysis show an uptick in Toastmasters growth and a pattern of continuing diversity.”

 

The infographic says there are 272,000+ members – a growth of 2.2%. But it doesn’t say how many clubs there are. So, I went to the Statistics and Data Hub and looked up the Fact Sheet for July 1, 2023 to June 30, 2024 which informed me that there were >13,800 clubs for a growth rate of -3.0%. What happened when the pandemic hit? Regular members don’t get told, but officers can find out. I looked up the CEO Reports for 2022 and 2024 and found the following data:

 

Year    Members   # of Clubs 

2018   357,718       16,672

2019   358,078       16,856

2020   364,212       16,204

2021   300,206       15,875

2022   282,055       14,749

2023   266,564       14,271

2024   272,338       13,846

   

From 2020 to 2021 membership dropped by 64,006 or 17.6%! And from 2019 to 2020 the number of clubs dropped by 652, or 3.9%. I found a chart showing Toastmaster International Membership Growth (1990 to 2020) so I added the data tabulated above, and used Excel to produce the chart for this century, from 2000 to 2024 shown at the top of this post. Membership now is almost at the same level as back in 2011.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I could not find a similar chart for the number of clubs, and looked up data from various web pages at the Toastmasters web site like one for 2011 and another for 2014. The chart shown above summarizes the situation from 2007 to 2024. In 2023 we had fallen back to barely above where we were a decade earlier in 2013. Note that multiplying the number of clubs (13,846 for 2024) by 20 members/club gives 276,920, which is pretty close to the membership of 272,338.        

  

Our organization was significantly wounded by the pandemic. Knowing that, we can work on recovering.  

 

A ludicrously overstated claim that Chapman University found 91% of Americans fear public speaking

 

 

 

 

 

 

 

 

 

There is an article by Jannik Linder at Gitnux on July 17, 2024 titled Statistics reveal widespread public speaking fear among individuals globally which contains a claim that:

 

“91% of Americans fear public speaking, according to a Chapman University study.”

 

But that is nonsense. Annually (except for 2020) from 2014 to 2023 Chapman University did their Chapman Survey of American Fears. They asked a random sample of U. S. adults about their fears at four levels: Not Afraid, Slightly Afraid, Afraid, and Very Afraid. In their press releases they reported the sum of percentages for Very Afraid and Afraid, as is shown above on a bar chart made using Excel. Those nine values ranged from 23.3% (2017) to 34% (2022) with a mean of 27.9%. The claimed 91% is more than three times too high!

 

 

 

 

 

 

 

 

 

What if we added the percent for Slightly Afraid? On October 23, 2023 I blogged about What do American adults fear at some level? Results for the Grand Sum of Very Afraid + Afraid + Slightly Afraid levels (everything but the kitchen sink) in the ninth 2023 Chapman Survey of American Fears. Public speaking was ranked #44 at 65.0%. Results for all nine surveys are shown above in a second bar chart. They range from 57.9% (2017) to 69.2% (2022) with a mean of 62%. Compared with the mean, that 91% is 29% too high!

  

Another problem with this article is that although there is a long list of references they only contains links to home pages, like one for Chapman University.

 

In this article there also is a claim that:

 

“77% of people have anxiety about public speaking.”

 

On October 12, 2020 I blogged about Do 77% of Americans fear public speaking? No! That percentage described stage fright in Swedes who also had social anxiety disorder.

 

There is a second article, also by Jannik Linder at Gitnux and from July 17, 2024 titled New findings reveal staggering public speaking fears statistics impact globally. It doesn’t have the 91% claim, but does say:  

 

“Public speaking is the number one fear for 25.3% of people.”

 

If you look at my first bar chart, then you will see that this percentage really is a result from the 2014 Chapman Survey of American Fears!

 

Five of six British age groups are more confident about public speaking after COVID

 

 

 

 

 

 

 

 

 

 

There is an article by Jon Brady at The Daily Mail on October 22, 2023 with the 35-word title Gen Z in confidence crisis: Third of 18-24-year-olds are too scared for public speaking or doing a presentation in front of fellow staff … after years of Zoom schooling during the pandemic. It contains results from a survey (done for Speak with Impact) which I have replotted above as a horizontal bar chart using Excel. Only the 18-to-24-year-old (Gen Z) were less confident. The other five age groups were more confident, so why weren’t they emphasized by Mr. Brady? There also was an article by Richard Eldred at The Daily Sceptic on October 22, 2023 titled Gen Zers Are Too Scared for Public Speaking …After Years of Zoom Schooling During the Pandemic.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Even worse, the bar chart used in both the articles (shown above) is badly flawed. First, and most obviously there is little contrast between the blue bars and the background. Second, pink is being used to represent More confident, and blue to represent Less confident – which is backward from what might be expected. Third, the percent More confident should be shown first, to the left of Less confident, since it is larger in five of six cases.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Whoever created that graphic deserves a visit and a scolding from the mythical Bar Chart Police.

 

 

 

 

 

 

 

 

 

 

 

How about the overall results? They were presented at Speak with Impact on August 2, 2023 in an article titled Impact of Covid on Presentation Confidence. As shown above, the majority (54%) were neither more mor less confident after the pandemic than before. 11% were much or slightly more confident, 26% were much or slightly less confident, and the other 8% didn’t know.

 

The Bar Chart Police badge was derived from this one at Openclipart.