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Histograms

6 min readโ€ขnovember 23, 2021

William

William

William

William

Histograms & Statistics

There are various ways to display data in statistics: box plots, stem-and-leaf plots, pie charts, and the like. Histograms are among the more popular ways to visualize data! Whether youโ€™re here to get some information about histograms for the first time or if you want more time to explore histograms as a statistician, this article is for you. ๐Ÿ“ฐ

Letโ€™s get started! ๐Ÿ“Š

What are Histograms?

Histograms are a popular method of showing numerical data in equal bins, otherwise known as bars that each hold a range of values. Each bin has two numbers, the first being inclusive and the second being exclusive. ๐Ÿ—‘๏ธ

The values in a histogram must comply with the standards of a condition, meaning they must be greater than or equal to the inclusive number and less than the second (exclusive) value. Unlike bar charts, which you might be more familiar with, histograms have bars that touch each other. If there's a gap in the data, there are no values that belong in that range of values, thus leaving an empty bin. ๐Ÿ˜ฑ

One-Variable Quantitative Data 1

Histograms are quantitative data with one variable. ๐Ÿ˜Œ

Wait, what does that mean? ๐Ÿ˜ณ

  • Quantitative data is data that can be measured or counted numerically. ๐Ÿงฎ

  • One-variable data only has one variable (in the example above, this one variable would be trees).ย 

The x-axis always contains the measured information, while the y-axis has the frequency (the number of occurrences that the measured data fits within certain conditionals).ย 

When creating a histogram with data, you want to ensure thatโ€ฆ

  1. The bins in the x-axis have equal bin ranges (gotta be consistent)!ย 

  2. The frequency number in the y-axis may not start at or increase by the same values as those on the x-axis!

  3. They should increase by an equal increment!

Now that weโ€™ve uncovered some features of histograms, letโ€™s take a look at an example histogram to explore this idea of bins further. ๐Ÿ’ก

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-E7FPZBZzFWog.png?alt=media&token=abba4def-a5c6-4c99-a62e-d41c40530dae

Image courtesy of Wiktionary

The histogram above has taken 31 black cherry trees and sorted them into bins based on their height. The first bin holds 3 black cherry trees with a height of at least 60 feet and less than 65 feet. We might also notice that the tallest bin of data with 10 trees requires heights of at least 75 feet but fewer than 80 feet. ๐ŸŒณ

Notice how the x-axisโ€™ left-most value does not start at 0. You can begin with any number relevant to the situation, but be aware that each subsequent tick should be equally further away so that each bin will only hold the same number of values. โœ”๏ธ

As for the y-axis, notice that the value at the x- and y- axes intersection is 0. Since the number of occurrences starts from 0 and increases, your y-axis will start with this value (for the most part) and increase in increments of an appropriate value. ๐Ÿ“ˆ

That's quite the good start for observations. Let's take our detective skills to the next level by making more concrete histogram interpretations! ๐Ÿ•ต๏ธ

Interpreting Histograms

Whether youโ€™re here to study for AP Statistics beyond Fiveableโ€™s AP resources or want to learn how to interpret histograms in your daily life thoroughly, this skill is handy to know. It will help you analyze data at the level of that of a statistician! ๐Ÿ˜บ

Four of the key elements you can identify when interpreting histograms are the Shape, any Unusual features, the Center, and the Spread (aka SUCS). Alternatively, you can use SOCS, where O stands for โ€œOutliers,โ€ which has an equivalent definition to Unusual features. ๐Ÿ”‘

Shape

Analyzing the shape of the graph requires taking a look at the histogram as a whole through a "birds-eye" view. ๐Ÿฆ

  • What's the rough shape of the curve all of the bars make? โญ•

  • Is there any upwards or downward trend? โ†•๏ธ

The histogram can also either be symmetrical or asymmetrical. Symmetry in a histogram doesn't necessarily mean that the histogram has bars that form a perfect curve or are each the same value; instead, if you were to draw a curve over the bars to show a trend, there is a general bell curve. ๐Ÿ””

Asymmetry is when the two sides of a histogram are not near-mirror images of each other and do not generally reflect each other. Asymmetry can come in the form of skewness, or a clear upwards or downwards trend that extends far outwards on one side of the histogram. ๐Ÿ‘Ž

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-xkjNGyIEthsg.png?alt=media&token=d2c2d4cf-ecd3-40a4-be53-f56fd4caf87e

Image courtesy of Wikipedia

Unusual Features (Outliers)

This analysis is typically observable; if you were to look at the graph, you would notice data that seems inconsistent with the histogram. While a high-frequency bar might look unusual compared to other bars, unusual features only encompass potential outliers and significant gaps between histogram bars. ๐Ÿคญ

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1ZBdhRbb3XW6.PNG?alt=media&token=dbade501-482d-4ca4-9614-72c30141b562

Center & Spread

The center of the data is where the data's mean or the median lies. By identifying any outliers or skewness, you can also discover the overall spread - statisticians call this โ€œmodalityโ€ - of the histogram, too! (For more information expanding on this topic, visit this AP Statistics study guide!) ๐ŸŽฏ

Practice: Histogram Interpretation

Letโ€™s put your skills to the test with a sample histogram from Wikipedia! You can find the answer at the bottom of this page. (Hint: Use SUCS!) ๐Ÿ’ช

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ZB0mFPPxRp2B.png?alt=media&token=b82d4f9c-26c3-42c3-b415-31880ff31b75

Image courtesy of Wikipedia

Histogram Interpretation Example Response

There are several ways you could have formulated your response, but here is an example that you can use! While this is a very short response, it still fulfills each of the points in the SUCS acronym! โซ

This histogram is (Spread) unimodal and (Shape) symmetric, with no visible (Unusual Features) potential outliers or gaps. (Center) The mean of the data lies between -0.5 and 0.5. โœŽ

Reflections: Pros & Cons of Histograms

While histograms are very useful at capturing data for visualizing trends or other means of data analysis, there are reasons that statisticians would instead use another type of graph. ๐Ÿ™…โ€โ™‚๏ธ

Pros ๐Ÿ˜

Cons & Limitations ๐Ÿ˜ 

  • Helps summarize large masses of data

  • Brings insight to occurrences distributions

  • Easy to visualize and and analyze trends

  • Categorizes numerical values easily into respective bins

  • Only able to capture data with one variable

  • Only able to use quantitative data

  • Cannot identify individual pieces of data from graph itself

  • Cannot calculate an exact mean, median, or mode

If you need to summarize a large amount of data, show a visual representation of data distribution, or want to categorize quantitative numbers into bins, you can do so with histograms! And with that saidโ€ฆ Incredible work on discovering all the features of a histogram; youโ€™re another step closer to becoming a statistician! Good luck on your studies, and see you next time. ๐ŸŒŠ

๐ŸคConnect with other students studying histograms withย Hours.

Histograms

6 min readโ€ขnovember 23, 2021

William

William

William

William

Histograms & Statistics

There are various ways to display data in statistics: box plots, stem-and-leaf plots, pie charts, and the like. Histograms are among the more popular ways to visualize data! Whether youโ€™re here to get some information about histograms for the first time or if you want more time to explore histograms as a statistician, this article is for you. ๐Ÿ“ฐ

Letโ€™s get started! ๐Ÿ“Š

What are Histograms?

Histograms are a popular method of showing numerical data in equal bins, otherwise known as bars that each hold a range of values. Each bin has two numbers, the first being inclusive and the second being exclusive. ๐Ÿ—‘๏ธ

The values in a histogram must comply with the standards of a condition, meaning they must be greater than or equal to the inclusive number and less than the second (exclusive) value. Unlike bar charts, which you might be more familiar with, histograms have bars that touch each other. If there's a gap in the data, there are no values that belong in that range of values, thus leaving an empty bin. ๐Ÿ˜ฑ

One-Variable Quantitative Data 1

Histograms are quantitative data with one variable. ๐Ÿ˜Œ

Wait, what does that mean? ๐Ÿ˜ณ

  • Quantitative data is data that can be measured or counted numerically. ๐Ÿงฎ

  • One-variable data only has one variable (in the example above, this one variable would be trees).ย 

The x-axis always contains the measured information, while the y-axis has the frequency (the number of occurrences that the measured data fits within certain conditionals).ย 

When creating a histogram with data, you want to ensure thatโ€ฆ

  1. The bins in the x-axis have equal bin ranges (gotta be consistent)!ย 

  2. The frequency number in the y-axis may not start at or increase by the same values as those on the x-axis!

  3. They should increase by an equal increment!

Now that weโ€™ve uncovered some features of histograms, letโ€™s take a look at an example histogram to explore this idea of bins further. ๐Ÿ’ก

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-E7FPZBZzFWog.png?alt=media&token=abba4def-a5c6-4c99-a62e-d41c40530dae

Image courtesy of Wiktionary

The histogram above has taken 31 black cherry trees and sorted them into bins based on their height. The first bin holds 3 black cherry trees with a height of at least 60 feet and less than 65 feet. We might also notice that the tallest bin of data with 10 trees requires heights of at least 75 feet but fewer than 80 feet. ๐ŸŒณ

Notice how the x-axisโ€™ left-most value does not start at 0. You can begin with any number relevant to the situation, but be aware that each subsequent tick should be equally further away so that each bin will only hold the same number of values. โœ”๏ธ

As for the y-axis, notice that the value at the x- and y- axes intersection is 0. Since the number of occurrences starts from 0 and increases, your y-axis will start with this value (for the most part) and increase in increments of an appropriate value. ๐Ÿ“ˆ

That's quite the good start for observations. Let's take our detective skills to the next level by making more concrete histogram interpretations! ๐Ÿ•ต๏ธ

Interpreting Histograms

Whether youโ€™re here to study for AP Statistics beyond Fiveableโ€™s AP resources or want to learn how to interpret histograms in your daily life thoroughly, this skill is handy to know. It will help you analyze data at the level of that of a statistician! ๐Ÿ˜บ

Four of the key elements you can identify when interpreting histograms are the Shape, any Unusual features, the Center, and the Spread (aka SUCS). Alternatively, you can use SOCS, where O stands for โ€œOutliers,โ€ which has an equivalent definition to Unusual features. ๐Ÿ”‘

Shape

Analyzing the shape of the graph requires taking a look at the histogram as a whole through a "birds-eye" view. ๐Ÿฆ

  • What's the rough shape of the curve all of the bars make? โญ•

  • Is there any upwards or downward trend? โ†•๏ธ

The histogram can also either be symmetrical or asymmetrical. Symmetry in a histogram doesn't necessarily mean that the histogram has bars that form a perfect curve or are each the same value; instead, if you were to draw a curve over the bars to show a trend, there is a general bell curve. ๐Ÿ””

Asymmetry is when the two sides of a histogram are not near-mirror images of each other and do not generally reflect each other. Asymmetry can come in the form of skewness, or a clear upwards or downwards trend that extends far outwards on one side of the histogram. ๐Ÿ‘Ž

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-xkjNGyIEthsg.png?alt=media&token=d2c2d4cf-ecd3-40a4-be53-f56fd4caf87e

Image courtesy of Wikipedia

Unusual Features (Outliers)

This analysis is typically observable; if you were to look at the graph, you would notice data that seems inconsistent with the histogram. While a high-frequency bar might look unusual compared to other bars, unusual features only encompass potential outliers and significant gaps between histogram bars. ๐Ÿคญ

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1ZBdhRbb3XW6.PNG?alt=media&token=dbade501-482d-4ca4-9614-72c30141b562

Center & Spread

The center of the data is where the data's mean or the median lies. By identifying any outliers or skewness, you can also discover the overall spread - statisticians call this โ€œmodalityโ€ - of the histogram, too! (For more information expanding on this topic, visit this AP Statistics study guide!) ๐ŸŽฏ

Practice: Histogram Interpretation

Letโ€™s put your skills to the test with a sample histogram from Wikipedia! You can find the answer at the bottom of this page. (Hint: Use SUCS!) ๐Ÿ’ช

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ZB0mFPPxRp2B.png?alt=media&token=b82d4f9c-26c3-42c3-b415-31880ff31b75

Image courtesy of Wikipedia

Histogram Interpretation Example Response

There are several ways you could have formulated your response, but here is an example that you can use! While this is a very short response, it still fulfills each of the points in the SUCS acronym! โซ

This histogram is (Spread) unimodal and (Shape) symmetric, with no visible (Unusual Features) potential outliers or gaps. (Center) The mean of the data lies between -0.5 and 0.5. โœŽ

Reflections: Pros & Cons of Histograms

While histograms are very useful at capturing data for visualizing trends or other means of data analysis, there are reasons that statisticians would instead use another type of graph. ๐Ÿ™…โ€โ™‚๏ธ

Pros ๐Ÿ˜

Cons & Limitations ๐Ÿ˜ 

  • Helps summarize large masses of data

  • Brings insight to occurrences distributions

  • Easy to visualize and and analyze trends

  • Categorizes numerical values easily into respective bins

  • Only able to capture data with one variable

  • Only able to use quantitative data

  • Cannot identify individual pieces of data from graph itself

  • Cannot calculate an exact mean, median, or mode

If you need to summarize a large amount of data, show a visual representation of data distribution, or want to categorize quantitative numbers into bins, you can do so with histograms! And with that saidโ€ฆ Incredible work on discovering all the features of a histogram; youโ€™re another step closer to becoming a statistician! Good luck on your studies, and see you next time. ๐ŸŒŠ

๐ŸคConnect with other students studying histograms withย Hours.



ยฉ 2024 Fiveable Inc. All rights reserved.

APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


ยฉ 2024 Fiveable Inc. All rights reserved.

APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.