Math Academy

Introduction

A histogram is a visual representation of a frequency table for grouped data.

For example, consider the data set below:

In this data set,

there is number between and
there are numbers between and
there are numbers between and and
there is number between and

We can organize these results in the following frequency table.

Value	Frequency

To draw the histogram, we divide a horizontal axis into equal subintervals, one for each range of values. Then, on each subinterval, we draw a rectangle whose height is equal to the frequency in the corresponding range.

The resulting graph is shown below:

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Example: Filling a Missing Column in a Frequency Table

The histogram below shows the height distribution, to the nearest centimeter, of a group of high school baseball players.

The frequency table below represents the same information. From left to right, what is missing from the table?

Height (cm)	Number of Players

EXPLANATION

The second column of the histogram tells us that there are players whose height is in the range of

Therefore, the missing parts are and

Height (cm)	Number of Players

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Practice: Filling a Missing Column in a Frequency Table

The histogram above shows the distribution of daily incomes, in US dollars, of the workers in a construction company.

The frequency table below represents the same information. From left to right, what is missing from the table?

Daily Income ($)	Number of Workers
$\fbox{$\,\phantom{000{-}000}\,$}$	$\fbox{$\,\phantom{0}\,$}$
$131{-}150$	$8$
$151{-}170$	$5$
$171{-}190$	$3$

a	$111{-}130$ and $6$
b	$101{-}120$ and $6.5$
c	$111{-}120$ and $8$
d	$131{-}150$ and $8$
e	$151{-}170$ and $5$

EXPLANATION

The first column of the histogram tells us that $\color{blue}6$ workers in the company have a daily income in the range $\color{blue}\$111{-}\$130.$

Therefore, the missing parts are $\color{blue}111{-}130$ and $\color{blue}6.$

Daily Income ($)	Number of Workers
$\fbox{$\, {\color{blue} 111{-}130}\,$}$	$\fbox{$\,{\color{blue} 6}\,$}$
$131{-}150$	$8$
$151{-}170$	$5$
$171{-}190$	$3$

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Practice: Filling a Missing Column in a Frequency Table

The histogram above shows the weight distribution of the vehicles loaded onto a ferry, measured to the nearest $0.1$ tons.

The frequency table below represents the same information. From left to right, what is missing from the table?

Weight (tons)	Number of Vehicles
$2.0{-}3.9$	$15$
$\fbox{$\,\phantom{0.0-0.0}\,$}$	$\fbox{$\,\phantom{0}\,$}$
$6.0{-}7.9$	$10$

a	$4.0{-}5.9$ and $5$
b	$2.0{-}3.9$ and $15$
c	$4.1{-}6.0$ and $6$
d	$4.0{-}7.9$ and $5$
e	$6.0{-}7.9$ and $10$

EXPLANATION

The second column of the histogram tells us that $\color{blue}5$ vehicles whose weight is in the range of $\color{blue}4.0{-}5.9$ $\text{tons}.$

Therefore, the missing parts are $\color{blue}4.0{-}5.9$ and $\color{blue}5.$

Weight (tons)	Number of Vehicles
$2.0{-}3.9$	$15$
$\fbox{$\,\color{blue}4.0{-}5.9\,$}$	$\fbox{$\,\color{blue}5\,$}$
$6.0{-}7.9$	$10$

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Example: Creating a Histogram From a Frequency Table

The frequency table below gives the weights, measured to the nearest kilogram, of some students in a statistics class.

Weight (kg)	Number of Students

Draw a histogram that represents the same data set.

EXPLANATION

From the frequency table:

students had weights in the range kilograms,
students had weights in the range kilograms,
students had weights in the range kilograms, and
students had weights in the range kilograms.

The histogram that represents this information is the following:

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Practice: Creating a Histogram From a Frequency Table

The frequency table below gives the number of calls handled per operator per day in a call center.

Number of Calls	Number of Operators
$1{-}10$	$10$
$11{-}20$	$25$
$21{-}30$	$20$
$31{-}40$	$15$

What histogram represents the same data set?

a
b
c
d
e

EXPLANATION

From the frequency table:

$\color{blue}10$ operators handled $1{-}10$ calls,

$\color{blue}25$ operators handled $11{-}20$ calls,

$\color{blue}20$ operators handled $21{-}30$ calls,

$\color{blue}15$ operators handled $31{-}40$ calls.

The histogram that represents this information is the following:

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Practice: Creating a Histogram From a Frequency Table

The frequency table below gives the number of customers per day for a small restaurant over the last month.

Number of Customers	Number of Days
$1{-}20$	$5$
$21{-}40$	$15$
$41{-}60$	$10$

What histogram represents the same data set?

a
b
c
d
e

EXPLANATION

From the frequency table:

on $\color{blue}5$ days, there were $1{-}20$ customers,

on $\color{blue}15$ days, there were $21{-}40$ customers,

on $\color{blue}10$ days, there were $41{-}60$ customers.

The histogram that represents this information is the following:

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Example: Finding a Partial Count From a Histogram

Tina sorted out her collection of science fiction books into four groups according to their number of pages. The histogram below gives the corresponding distribution. How many of Tina's science fiction books have at most pages?

EXPLANATION

The first and second bars correspond to the books that have between and pages, respectively.

The height of the first bar is (halfway between and ) and the second bar has a height of (halfway between and ).

Therefore, Tina has books that have at most pages.

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Practice: Finding a Partial Count From a Histogram

Brad, a baseball trainer, formed five groups of children according to their ages. The histogram above gives the corresponding distribution. How many of the children are at least $11$ years old?

a	$5$
b	$2$
c	$8$
d	$6$
e	$9$

EXPLANATION

The fourth and fifth bars correspond to the children aged between $\color{blue}11{-}12$ and $\color{blue} 13{-}14$ years old, respectively.

The height of the fourth bar is $\color{red}3$ (halfway between $2$ and $4$ ), and the fifth bar has a height of $\color{red}5$ (halfway between $4$ and $6$ ).

Therefore, there were ${\color{red}3} + {\color{red}5} = 8$ children that are at least $11$ years old.

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Practice: Finding a Partial Count From a Histogram

As part of a game on Sharon's Birthday, her friends were divided into three groups according to their heights. The histogram above gives the corresponding distribution. How many of Sharon's friends are shorter than $1.7$ meters?

a	$8$
b	$17$
c	$9$
d	$12$
e	$13$

EXPLANATION

The first and second bars correspond to those whose height is between $\color{blue} 1.3{-}1.4$ and $\color{blue} 1.5{-}1.6$ meters, respectively.

The height of the first bar is $\color{red}8$ , and the second bar has a height of $\color{red}9$ (halfway between $8$ and $10$ ).

Therefore, there were ${\color{red}8} + {\color{red}9} = 17$ people shorter than $1.7$ meters.

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Tails, Skew, and Symmetry

A tail of a distribution is a part that extends away from the main cluster.

If the left tail of the distribution is longer than the right tail, then we say that the distribution is left-skewed (or negatively skewed). An example of a left-skewed distribution is shown below.

Similarly, if the right tail of the distribution is longer than the left tail, then we say that the distribution is right-skewed (or positively skewed). An example of a right-skewed distribution is shown below.

If the distribution's tails are the same, we say that the distribution is symmetric. An example of symmetric distribution is shown below.

For symmetric distributions, the mean and the median always lie in the middle group. So, in this case, we know that the mean and median both lie within the values and

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The Mean and Median for Symmetric Distributions, and Modal Class

When a distribution is symmetric, and the corresponding histogram has an odd number of bars, the mean and the median always lie within the central column. So, in the case below, both the mean and median lie between and

When a distribution is symmetric, and the corresponding histogram has an even number of bars, the mean and the median always lie within the two central columns. So, in the case below, both the mean and median lie between and

Note that it's usually impossible to precisely determine the mean and median from a histogram alone since the data is grouped. We can only estimate.

Finally, for any set of grouped data, the modal class is the class (i.e., group) corresponding to the tallest bar on the histogram. For the histogram below, the modal class is

Similar to the mode, some data sets have more than one modal class. For example, the symmetric graph with six classes shown above (the one with two blue bars) has two modal classes, and

Other data sets have no modal class:

If each class has a height of then there is no modal class.
Similarly, if each class has the same height, then there is no modal class.

In both cases, we would have a completely flat histogram.

For example, the data set below has no modal class.

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Example: Identifying Characteristics of Distributions Using Histograms

Members of a gym were asked how much time they spend working out on a typical day. The histogram below shows the results. Which of the following statements are true?

The distribution is left-skewed.
The distribution is symmetric.
The distribution is right-skewed.
The mean lies between and minutes.

EXPLANATION

Let's look at the statements one by one.

Statement II is true, while statements I and III are false. The left-hand side is a mirror image of the right-hand side. Therefore, the distribution is symmetric.

Statement IV is false. When a distribution is symmetric, and the corresponding histogram has an odd number of bars, the mean and the median always lie within the central column. Therefore, the mean time lies between and minutes.

Therefore, the correct answer is "II only."

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Practice: Identifying Characteristics of Distributions Using Histograms

The histogram above reports the times, in minutes, that visitors of an amusement park had to wait in line before entering the park. Which of the following statements are true?

The distribution is right-skewed.
The distribution is symmetric.
The distribution is left-skewed.
The modal class is $20{-}29$ minutes.

a	II only
b	III only
c	I and IV only
d	III and IV only
e	II and IV only

EXPLANATION

Let's examine the statements one by one.

Statement I is true, while statements II and III are false. The given histogram shows that the distribution has a long right tail. Therefore, it is right-skewed.

Statement IV is true. The tallest column corresponds to the range $\color{blue}20{-}29.$ Therefore, the modal class is $20{-}29$ minutes.

Therefore, the correct answer is "I and IV only."

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Practice: Identifying Characteristics of Distributions Using Histograms

A group of graduate students was asked how long they watch TV every day. The histogram above shows the corresponding data. Which of the following statements are true?

The distribution is symmetric.
The distribution is right-skewed.
The distribution is left-skewed.
The median time lies between $40$ and $59$ minutes.

a	I and IV only
b	II only
c	III and IV only
d	II and IV only
e	I only

EXPLANATION

Let's examine the statements one by one.

Statement I is true, while statements II and III are false. The left-hand side is a mirror image of the right-hand side. Therefore, the distribution is symmetric.

Statement IV is true. When a distribution is symmetric, and the corresponding histogram has an odd number of bars, the mean and the median always lie within the central column. Therefore, the median time lies between $40$ and $59$ minutes.

Therefore, the correct answer is "I and IV only."

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