How to Read a Frequeency Allocation Chart

Frequency Tabular array

A frequency tabular array is only a "t-nautical chart" or two-column table which outlines the diverse possible outcomes and the associated frequencies observed in a sample.

From: The Joy of Finite Mathematics , 2016

Describing Information Sets

Sheldon G. Ross , in Introductory Statistics (Quaternary Edition), 2017

2.2.1 Line Graphs, Bar Graphs, and Frequency Polygons

Information from a frequency table can be graphically pictured past a line graph, which plots the successive values on the horizontal centrality and indicates the corresponding frequency by the top of a vertical line. A line graph for the data of Table ii.1 is shown in Fig. ii.1.

Figure ii.i. A line graph.

Sometimes the frequencies are represented not by lines but rather by bars having some thickness. These graphs, called bar graphs, are often utilized. Figure ii.2 presents a bar graph for the data of Table 2.1.

Figure 2.two. A bar graph.

Another blazon of graph used to stand for a frequency tabular array is the frequency polygon, which plots the frequencies of the different data values and then connects the plotted points with directly lines. Figure 2.3 presents the frequency polygon of the data of Table 2.ane.

Figure 2.iii. A frequency polygon.

A set of data is said to be symmetric near the value ${\mathrm{10}}_0$ if the frequencies of the values $x_0-c$ and $x_0+c$ are the same for all c. That is, for every constant c, there are just every bit many data points that are c less than $x_0$ as there are that are c greater than ${\mathrm{10}}_0$ . The information set presented in Table 2.ii, a frequency tabular array, is symmetric about the value ${\mathrm{10}}_0=3$ .

Table 2.2. Frequency Table of a Symmetric Data Set

Value	Frequency	Value	Frequency
0	ane	four	2
2	2	6	1
3	3	0	0

Information that are "close to" being symmetric are said to be approximately symmetric. The easiest manner to make up one's mind whether a information set is approximately symmetric is to correspond it graphically. Figure two.4 presents three bar graphs: one of a symmetric data set, one of an approximately symmetric data set, and one of a information prepare that exhibits no symmetry.

Effigy 2.4. Bar graphs and symmetry.

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Descriptive Methods

Ronald N. Forthofer , ... Mike Hernandez , in Biostatistics (Second Edition), 2007

three.two.one Frequency Tables

A one-fashion frequency table shows the results of the tabulation of observations at each level of a variable. In Table 3.2, we show one-way tabulations of sex and race for the xl patients shown in Tabular array three.1. 3 quarters of the patients are males, and over 87 pct of the patients are whites.

Table 3.2. Frequencies of sex and race for 40 patients in DIG40.

Sexual practice	Number of Patients	Percentage	Race	Number of Patients	Percent
Male	xxx	75.0	White	35	87.5
Female	ten	25.0	Nonwhite	5	12.v
Total	40	100.0	Total	40	100.0

The variables used in frequency tables may be nominal, ordinal, or continuous. When continuous variables are used in tables, their values are often grouped into categories. For example, age is often categorized into x-year intervals. Tabular array 3.three shows the frequencies of age groups for the forty patients in Tabular array 3.1. More than than one one-half of the patients are 60 and over. Annotation that the sum of percents should add up to 100 percent, although a pocket-size assart is made for rounding. It is too worth noting that the title of the table should contain sufficient data to allow the reader to sympathise the table.

Tabular array 3.iii. Frequency of age groups for 40 patients in DIG40.

Age Groups	Number of Patients	Percent
Under xl	3	7.v
40–49	vi	xv.0
l–59	8	20.0
60–69	11	27.5
lxx–79	12	30.0
Total	40	100.0

Ii-way frequency tables, formed past the cantankerous-tabulation of ii variables, are usually more interesting than one-manner tables because they show the relationship betwixt the variables. Tabular array 3.4 shows the relationship between sex and body mass index where BMI has been grouped into underweight (BMI < 18.5), normal (18.5 ≤ BMI < 25), overweight (25 ≤ BMI < 30), and obese (BMI ≥ 30). The body mass index is calculated as weight in kilograms divided by height in meters squared. There are higher percentages of females in the overweight and obese categories than those plant for males, but these calculations are based on very small sample sizes.

Table 3.4. Cantankerous-tabulation of torso mass alphabetize and sex for 40 patients in DIG40 with cavalcade percentages in parentheses.

	Sex
Body Mass Index	Male	Female person	Total
Under 18.5 (underweight)	1 (3.three%)	0 (0.0%)	one (ii.5%)
18.5–24.9 (normal)	x (33.3%)	2 (20.0%)	12 (30.0%)
25.0–29.9 (overweight)	14 (46.vii%)	6 (threescore.0%)	twenty (50.0%)
thirty.0 & over (obese)	5 (16.vii%)	2 (20.0%)	vii (17.v%)
Full	30	10	40

In forming groups from continuous variables, we should non allow the data to guide united states. We should use our knowledge of the subject matter, and non use the data, in determining the groupings. If we use the data to guide united states, information technology is easy to obtain apparent differences that are not existent but but artifacts. When we encounter categories with no or few observations, we tin can reduce the number of categories by combining or collapsing these categories into the next categories. For example, in Table 3.four the number of obesity levels can exist reduced to three past combining the underweight and normal categories. There is no need to subdivide the overweight category, even though one-one-half of observations are in this category. Estimator packages tin be used to categorize continuous variables (recoding) and to tabulate the data in one- or 2-mode tables (see Programme Note 3.1 on the website).

There are several ways of displaying the data in a tabular format. In Tables 3.2, 3.3 and three.4 Tables 3.2, 3.3, and 3.4 nosotros showed both numbers and percentages, just it is not necessary to show both in a summary table for presentation in periodical articles. Table 3.5 presents bones patient characteristics for 200 patients from the DIG200 data set. Note that the total number (due north) relevant to the percentages of each variable is presented at the pinnacle of the column and percentages alone are presented, leaving out the frequencies. The frequencies can exist calculated from the percentages and the total number.

Tabular array three.5. Basic patient characteristics at baseline in the Digoxin clinical trial based on 200 patents in DIG200.

Characteristics		Percentage (n = 200)
Sexual practice	Male person	73.0
	Female person	27.0
Race	White	86.five
	Nonwhite	13.five
Historic period	Under xl	three.five
	40–49	11.5
	50–59	25.0
	60–69	33.0
	70 & over	26.0
Body mass alphabetize	Underweight (< eighteen.five)	1.five
	Normal (18.5–24.9)	37.five
	Overweight (25–29.nine)	42.5
	Obese (≥ 30)	18.5

Other data besides frequencies tin can be presented in a tabular format. For example, Table 3.6 shows the health expenditures of three nations as a percentage of gross domestic products (Gdp) over time (NCHS 2004, Table 115). Health expenditures every bit a percent of GDP are increasing much more rapidly in the United States than either Canada or Uk.

Table three.six. Health expenditures as a per centum of gross domestic product over time.

Twelvemonth	Canada	United Kingdom	U.s.
1960	five.4	3.9	5.one
1965	5.6	4.1	vi.0
1970	seven.0	four.5	seven.0
1975	vii.0	5.five	8.4
1980	vii.1	v.vi	viii.8
1985	8.0	half-dozen.0	10.6
1990	9.0	6.0	12.0
1995	nine.2	7.0	xiii.iv
2000	9.ii	7.3	13.iii

Source: National Eye for Health Statistics, 2004, Tabular array 115

Copyright © 2004

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Bones Statistics

Chris Tsokos , Rebecca Wooten , in The Joy of Finite Mathematics, 2016

Basic Problems

8.two.five.

Outset construct a frequency table for the data then construct a bar nautical chart for the qualitative data. Construct a Pareto chart.

C	B	C	B	Due east	B	C	C	A	C
A	E	D	Eastward	A	A	B	C	C	D
B	D	A	B	D	D	E	B	D	A
East	A	D	B	C	D	C	C	E	A
B	D	B	D	B	C	A	D	A	C
A	E	A	D	A	A	A	Eastward	B	E

What useful data can be obtained past reading the bar chart and the Pareto nautical chart?

eight.2.6.

Beginning construct a frequency table for the given data obtained from tossing a die 25 times; and then construct a bar chart for the discrete data that represents the number of dots uppermost on a dice.

4	1	3	iii	5
2	three	iv	i	2
four	2	5	2	2
4	ane	five	two	3
1	1	half-dozen	6	one
1	3	4	4	1

Does this dice appear to be off-white? What would be expected for a fair dice?

eight.2.vii.

Given the following bar nautical chart for color of cars observed in a parking lot.

a.

Which color has the least amount of cars in the parking lot?

b.

How many more red cars are there than blue cars?

c.

How many cars are there in the parking lot?

d.

Construct the Pareto Chart.

e.

What is the relative frequency of blueish cars?

8.ii.8.

Given the following bar chart for points earned an a math test.

a.

Who had the highest grade?

b.

Which pupil(south) had a grade of a to the lowest degree 75 points?

8.2.ix.

Given the following bar chart for the number of minutes spent talking on the telephone by each child,

a.

How much longer did Andrew talk on the telephone than Kaitlin?

b.

Who was on the telephone for more than than 1-fourth of an hour?

eight.2.10.

Given the following pie chart for favorite flavor,

a.

If 300 individuals voted, how many people chose wintertime as their favorite season?

b.

What is the least popular season?

c.

Assuming 300 individuals voted, construct a bar chart using this data.

8.2.11.

Given the following pie chart for how students get to school,

a.

If the school has 600 students, how many students walk to school?

b.

What fraction of the students bulldoze to school?

8.2.12.

Given the following pie chart for how many books each pupil read this year,

a.

If 379 students were surveyed, how many students read exactly iii books?

b.

What fraction of the students read at least 3 books?

viii.two.13.

Given the following pie chart for what students drank for breakfast,

a.

What is the nigh pop beverage?

b.

What fraction of the students drank milk for breakfast?

Contingency tables

8.2.fourteen.

Consummate the following contingency table

8.2.15.

Complete the following contingency tabular array

eight.2.16.

Construct a contingency tabular array for the post-obit survey data:

Using the synthetic contingency table, answer the following questions:

a)

How many individuals where included in the survey?

b)

How many males? How many females?

c)

What is the nigh favored color?

d)

What is the least favored color?

due east)

How many individuals favor red?

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Accompaniment Tools for Doing Information Mining

Robert Nisbet Ph.D. , ... Ken Yale D.D.South., J.D. , in Handbook of Statistical Analysis and Data Mining Applications (2nd Edition), 2018

Frequency Tables

In practically every research projection, an initial examination of the data set usually includes frequency tables. In survey inquiry, for instance, frequency tables can show the number of males and females who participated in the survey, the number of respondents from particular indigenous and racial backgrounds, and so on. Responses on some labeled attitude measurement scales (e.1000., interest in watching football) tin also be nicely summarized via the frequency table. In medical research, you may tabulate the number of patients displaying specific symptoms; in industrial research, you may tabulate the frequency of dissimilar causes leading to catastrophic failure of products during stress tests (e.g., which parts are actually responsible for the complete malfunction of television sets under extreme temperatures). Customarily, if a data set includes any chiselled data, then one of the outset steps in the data analysis is to compute a frequency table for those categorical variables.

Frequency or one-manner tables represent the simplest method for analyzing categorical (nominal) data. They are used often to review how different categories of values are distributed in the sample. For example, in a survey of spectator interest in unlike sports, we could summarize the respondents' interest in watching football in a frequency table, as shown in Table half-dozen.1

Table 6.one. Frequency of Respondents' Interest in Watching Football Games

Table 6.i shows the number, proportion, and cumulative proportion of respondents who characterized their interest in watching football equally (1) ever interested, (2) usually interested, (three) sometimes interested, or (4) never interested.

Frequency tables tin also be tabulated for continuous data. In STATISTICA Information Miner, the Frequency Table function generates frequency tables and histograms for both continuous and categorical variables. Users tin specify the number of intervals for continuous variables. STATISTICA will automatically categorize categorical variables by codes if they are specified; otherwise, all distinct values in the categorical variables volition be identified. Users have control over 2 additional aspects of frequency tables: (1) type of categorization, where users specify the method of categorization for continuous variables (for categorical variables, either specific codes are used or all integer values are identified), and (2) number of intervals, where you can change the number of significant digits that are used when labeling the category levels in the graph by specifying the desired number of intervals.

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Using Statistics to Summarize Information Sets

Sheldon Thou. Ross , in Introductory Statistics (Third Edition), 2010

Solution

Since the original data set consists of the half-dozen values

$3,3,4,5,5,5$

information technology follows that the sample mean is

$\begin{array}{ll}\overline{x}\hfill & =\frac{3+\mathrm{three}+4+5+5+\mathrm{five}}{\mathrm{vi}}\hfill \\ \hfill & =\frac{3\times 2+4\times 1+5\times 3}{\mathrm{half-dozen}}\hfill \\ \hfill & =\frac{25}{6}\hfill \end{array}$

That is, the sample mean of the number of suits sold daily is 4.25.

In Example 3.3 nosotros take seen that when the information are arranged in a frequency table, the sample mean tin be expressed as the sum of the products of the distinct values and their frequencies, all divided by the size of the data set. This event holds in full general. To see this, suppose the information are given in a frequency table that lists thousand distinct values x ₁, x ₂, …, ten_{one thousand} with respective frequencies f ₁, f ₂, …, f_k . It follows that the data set consists of n observations, where $n={\sum }_{i=1}^{\mathrm{one\; thousand}}{f}_i$ and where the value x_i appears f_i times for i = 1, two, …, k. Hence, the sample mean for this data ready is

(3.1) $\begin{array}{ll}\overline{x}\hfill & =\frac{x_1+\dots +x_1+{\mathrm{10}}_2+\dots +x_{\mathrm{ii}}+\dots +x_m+\dots +{\mathrm{ten}}_{\mathrm{one\; thousand}}}{n}\hfill \\ \hfill & =\frac{f_1{\mathrm{10}}_1+f_2{\mathrm{10}}_2+\dots +f_k{\mathrm{10}}_k}{\mathrm{north}}\hfill \end{array}$

At present, if w _ane, w ₂, …, w_k are nonnegative numbers that sum to 1, then

$w_1{\mathrm{10}}_1+{\mathrm{due\; west}}_{\mathrm{ii}}{x}_2+\dots +{\mathrm{westward}}_k{\mathrm{10}}_k$

is said to be a weighted average of the values 10 _i, x ₂, …, x_grand with west_i beingness the weight of ten_i . For instance, suppose that k = 2. At present, if w _one = w ₂ = ane/ii, then the weighted boilerplate

${\mathrm{west}}_1{x}_1+{\mathrm{westward}}_2{x}_2=\frac{1}{2}{x}_{\mathrm{i}}+\frac{1}{\mathrm{two}}{x}_{\mathrm{two}}$

is just the ordinary average of x _one and x ₂. On the other hand, if w ₁ = two/iii and w _two = 1/3, then the weighted average

${\mathrm{due\; west}}_{\mathrm{one}}{\mathrm{10}}_1+w_2{x}_2=\frac{2}{\mathrm{iii}}{\mathrm{ten}}_1+\frac{1}{3}{x}_{\mathrm{two}}$

gives twice as much weight to x ₁ equally it does to x ₂.

By writing Eq. (3.1) as

$\overline{x}=\frac{f_1}{\mathrm{north}}{\mathrm{10}}_1+\frac{f_{\mathrm{two}}}{n}{x}_{\mathrm{ii}}+\dots +\frac{f_k}{\mathrm{north}}{x}_{\mathrm{grand}}$

we see that the sample mean

is a weighted average of the fix of distinct values. The weight given to the value x_i is f_i /northward, the proportion of the information values that is equal to x_i. Thus, for instance, in Example 3.3 we could have written that

$\overline{x}=\frac{2}{6}\times \mathrm{iii}+\frac{1}{\mathrm{six}}\times 4+\frac{3}{\mathrm{vi}}\times 5=\frac{25}{\mathrm{half\; dozen}}$

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Descriptive statistics

Kandethody M. Ramachandran , Chris P. Tsokos , in Mathematical Statistics with Applications in R (Third Edition), 2021

1.5.ane Numerical measures for grouped data

When nosotros meet situations where the data are grouped in the course of a frequency tabular array (see Section 1.4), nosotros no longer have individual data values. Hence, nosotros cannot utilise the formulas in Definition i.7.ane. The post-obit formulas will requite estimate values for $\overline{\mathrm{10}}$ and s ². Let the grouped data have 50 classes, with m _i being the midpoint and f _i beingness the frequency of grade i, i  =   1, ii, …, fifty. Let $n=\sum _{i=1}^l{f}_i$ .

Definition i.v.4

The mean for a sample of size n,

$\overline{\mathrm{10}}=\frac{1}{\mathrm{northward}}\sum _{i=1}^{\mathrm{50}}{f}_i{m}_i,$

where grand_i is the midpoint of the form i and f_i is the frequency of the class i.

Similarly, the sample variance ,

${\mathrm{southward}}^{\mathrm{ii}}=\frac{1}{n-1}\sum _{i=1}^n{f}_i{(m_i-\overline{\mathrm{ten}})}^{\mathrm{two}}=\frac{\sum g_i^2{f}_i-\frac{{\left(\sum _i{f}_i{\mathrm{thousand}}_i\right)}^2}{n}}{n-\mathrm{i}}.$

The post-obit example illustrates how we calculate the sample mean for a grouped information.

Example i.five.iv

The grouped data in Table ane.18 represent the number of children from nativity through the end of the teenage years in a large flat circuitous. Find the mean, variance, and standard deviation for these information.

Table 1.18. Number of Children and Their Age Grouping.

Class	0–3	4–7	8–11	12–fifteen	16–nineteen
Frequency	seven	4	19	12	eight

Here we utilize the usual convention of until the child attains the adjacent age, the age will be the previous year, for instance until a child is 4   years one-time, we volition say the kid is iii   years quondam.

Solution

Note that even though the classes are given as disjoint, in actuality these are next age intervals, like [0, iv), [iv, viii), etc. When we take the class midpoint, we have to have this into account. For simplicity of calculation we create Table 1.xix .

Tabular array 1.xix. Summary Statistics for Number of Children.

Class	Interval	f i	grand i	m i f i	${\mathit{m}}_{\mathit{i}}^2{\mathit{f}}_{\mathit{i}}$
0–iii	[0, 4)	7	2	xiv	28
4–7	[four, 8)	4	vi	24	144
8–xi	[8, 12)	19	10	190	1900
12–15	[12, 16)	12	fourteen	168	2352
16–xix	[16, xx)	8	eighteen	144	2592
	n  =   50		$\sum {\mathrm{yard}}_i{f}_i=540$	$\sum m_i^2{f}_i=7016$

The sample mean is

$\overline{x}=\frac{1}{n}\sum _i{f}_i{m}_i=\frac{540}{\mathrm{fifty}}=\mathrm{10.fourscore.}$

The sample variance is

$s^2=\frac{\sum m_i^2{f}_i-\frac{{\left(\sum _i{f}_i{m}_i\right)}^2}{n}}{\mathrm{north}-1}=\frac{7016-\frac{{\left(540\right)}^{\mathrm{ii}}}{50}}{49}=24.1632650\approx \mathrm{24.xvi.}$

The sample standard deviation is $\mathrm{southward}=\sqrt{{\mathrm{southward}}^2}=4.9156144\approx \mathrm{4.92.}$

Using the following calculations, we tin also detect the median for grouped data. We only know that the median occurs in a item grade interval, but nosotros practise not know the exact location of the median. We will presume that the measures are spread evenly throughout this interval. Let

L   = lower class limit of the interval that contains the median.

northward   = total frequency.

F _b   = cumulative frequencies for all classes earlier the median grade.

f _m   = frequency of the class interval containing the median.

due west   = interval width of the interval that contains the median.

Then the median for the grouped data is given by

$\mathrm{One\; thousand}=\mathrm{50}+\frac{w}{f_{\mathrm{grand}}}\left(\mathrm{0.v}n-F_b\right).$

We proceed to illustrate with an case.

Case one.5.5

For the data in Example 1.5.4, notice the median.

Solution

Get-go, nosotros develop Table 1.20.

Table 1.20. Frequency Distribution for Number of Children.

Class	f i	Cumulative f i	Cumulative f i /n
0–3	vii	seven	0.14
four–7	4	11	0.22
viii–11	19	30	0.vi
12–xv	12	42	0.84
16–19	8	fifty	one.00

The first interval for which the cumulative relative frequency exceeds 0.5 is the interval that contains the median. Hence, the interval 8 to 11 contains the median. Therefore, Fifty  =   8, f _chiliad   =   19, n =   l, w =   iii, and F _b   =   eleven. Then, the median is

$\mathrm{Thousand}=\mathrm{50}+\frac{w}{f_{\mathrm{thou}}}(0.5\mathrm{northward}-F_b)=8+\frac{3}{19}(\left(0.5\right)\left(50\right)-11)=\mathrm{10.211.}$

It is of import to note that all the numerical measures we calculate for grouped data are simply approximations to the actual values of the ungrouped data if they are available.

One of the uses of the sample standard deviation will exist clear from the following result, which is based on the data post-obit a bell-shaped curve. Such an indication can be obtained from the histogram or stem-and-leaf display.

Empirical rule

When the histogram of a data set is "bell-shaped" or "mound-shaped," and symmetric, the empirical rule states:

1.

Approximately 68% of the data are in the interval $\left(\overline{x}-\mathrm{due\; south},\overline{x}+s\right)$ .

2.

Approximately 95% of the data are in the interval $\left(\overline{x}-2s,\overline{x}+2\mathrm{south}\right)$ .

3.

Approximately 99.7% of the data are in the interval $\left(\overline{x}-3s,\overline{x}\mathrm{+}3s\right)$ .

The bell-shaped curve is called a normal curve and is discussed afterwards in Chapter 3. A typical symmetric bell-shaped bend is given by Fig. 1.v.

Effigy ane.5. Bell-shaped curve.

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Descriptive statistics

Sheldon M. Ross , in Introduction to Probability and Statistics for Engineers and Scientists (Sixth Edition), 2021

two.2.ane Frequency tables and graphs

A data set up having a relatively small number of distinct values can be conveniently presented in a frequency tabular array . For instance, Tabular array 2.1 is a frequency tabular array for a information gear up consisting of the starting yearly salaries (to the nearest thousand dollars) of 42 recently graduated students with B.South. degrees in electrical engineering. Table 2.1 tells us, amidst other things, that the lowest starting salary of $57,000 was received past four of the graduates, whereas the highest salary of $70,000 was received past a single educatee. The most common starting bacon was $62,000, and was received by 10 of the students.

Table 2.1. Starting Yearly Salaries.

Starting Salary	Frequency
57	four
58	i
59	iii
60	v
61	8
62	10
63	0
64	5
66	2
67	iii
lxx	1

Information from a frequency table tin can be graphically represented by a line graph that plots the singled-out data values on the horizontal axis and indicates their frequencies by the heights of vertical lines. A line graph of the information presented in Table 2.1 is shown in Figure 2.1.

Figure 2.1. Starting salary data.

When the lines in a line graph are given added thickness, the graph is called a bar graph. Effigy two.2 presents a bar graph.

Figure 2.2. Bar graph for starting salary information.

Another type of graph used to represent a frequency tabular array is the frequency polygon, which plots the frequencies of the different data values on the vertical axis, and then connects the plotted points with straight lines. Effigy two.three presents a frequency polygon for the data of Tabular array 2.1.

Figure 2.3. Frequency polygon for starting salary data.

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