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From Wikipedia

Midpoint

The midpoint (also known as class mark in relation to histogram) is the middle point of a line segment. It is equidistant from both endpoints.

Formulas

The formula for determining the midpoint of a segment in the plane, with endpoints (x1) and (x2) is:

\frac{x_1 + x_2}{2}

The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1) and (x2, y2) is:

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1, z1) and (x2, y2z2) is:

\left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)

More generally, for an n-dimensional space with axes x_1, x_2, x_3, \dots, x_n\,\!, the midpoint of an interval is given by:

\left(\frac{x_{1_1} + x_{1_2}}{2}, \frac{x_{2_1} + x_{2_2}}{2}, \frac{x_{3_1} + x_{3_2}}{2}, \dots , \frac{x_{n_1} + x_{n_2}}{2} \right)

Construction

The midpoint of a line segment can be located by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint. It is more challenging to locate the midpoint using only a compass, but it is still possible.


Frequency distribution

In statistics, a frequency distribution is a tabulation of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way the table summarizes the distribution of values in the sample.

Univariate frequency tables

Univariate frequency distributions are often presented as lists ordered by quantity showing the number of times each value appears. For example, if 100 people rate a five-point Likert scale assessing their agreement with a statement on a scale on which 1 denotes strong agreement and 5 strong disagreement, the frequency distribution of their responses might look like:

A different tabulation scheme aggregates values into bins such that each bin encompasses a range of values. For example, the heights of the students in a class could be organized into the following frequency table.

A Frequency Distribution shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data e.g. to show results of an election, income of people for a certain region, sales of a product within a certain period, student loan amounts of graduates, etc. Some of the graphs that can be used with frequency distributions are histograms, line graphs, bar charts and pie charts. Frequency distributions are used for both qualitative and quantitative data..

Joint frequency distributions

Bivariate joint frequency distributions are often presented as (two-way) contingency tables:

The total row and total column report the marginal frequencies or marginal distribution, while the body of the table reports the joint frequencies.

Applications

Managing and operating on frequency tabulated data is much simpler than operation on raw data. There are simple algorithms to calculate median, mean, standard deviation etc. from these tables.

Statistical hypothesis testing is founded on the assessment of differences and similarities between frequency distributions. This assessment involves measures of central tendency or averages, such as the mean and median, and measures of variability or statistical dispersion, such as the standard deviation or variance.

A frequency distribution is said to be skewed when its mean and median are different. The kurtosis of a frequency distribution is the concentration of scores at the mean, or how peaked the distribution appears if depicted graphically—for example, in a histogram. If the distribution is more peaked than the normal distribution it is said to be leptokurtic; if less peaked it is said to be platykurtic.

Letter frequency distributions are also used in frequency analysis to crack codes and refer to the relative frequency of letters in different languages.



From Yahoo Answers

Question:Hi, general question here about histograms. what does it show us at a glance? and how do you find YOUR value from the spread of the data displayed on there?

Answers:Remember you need a frequency table to draw a histogram. The table gives you the frequency distribution, how frequent values within a certain interval (group) occurs. Then you draw connecting rectangular charts, height = frequency, length = class interval, for each pair. At a glance, you can see which group of data happens more frequent, less frequent, ... and so on. If the data is about scores, then you can see which range of scores are obtained more often, less often. Your own score is included in one of the class intervals, but all scores in the interval are represented by its mid-point. Yours can be lower, larger, or even the mid-point itself. Is it ok?

Question:I need to make a histogram for statistics class and I can't figure out how to use the program StatPlus:mac (with Excel for mac) to make the histogram. Could someone tell me where to find instructions to make a histogram? or if there is any other way I can make a histogram on excel?

Answers:Find someone that has Microsoft Office installed on their PC and will let you use it. Open Microsoft Excel and go to "Tools", "Add-ins". Place a check in the box beside "Analysis ToolPak", click OK. Create a table with the following columns: Time of Day, Accidents, & Range. Enter the data in the following image. Go to "Tools", "Data Analysis" or "Data", "Data Analysis". Click on "Histogram" then click OK. Click the box beside "Input Range" and highlight the "Accidents" data. Click the box beside "Bin Range" and highlight the "Range" data. The Input Range is the raw data while the Bin Range provides the intervals to separate the data into. In the "Output Options" select "Output Range". Select an empty cell within the worksheet then press OK. This places the histogram and corresponding table on the same worksheet as the original information. According to the histogram, the most accidents occur at five hours during the day; when referring to the data table, those hours are 1am, 4am, 2pm (1400 hours), 5pm (1700 hours), and 11pm (2300 hours).

Question:When should I use a histogram instead of a polygon? Can any one explain when would it be better to use a histogram instead of a polygon and why? Please give an example. Ten points for the best answer. I know what a Histogram and polygon is, but I need an example of why a histogram is better.

Answers:A frequency polygon is used instead of a histogram when you wish to compare two sets of data. You can plot two polygons on the same axes and still identify which is which. That's because a polygon is like a line graph. If you plot two histograms on the same axes then it is difficult to distinguish between them. For that reason you can say that a frequency polygon is better than histogram. Have you been asked why a histogram is better than a frequency polygon? There is a complication with a frequency polygon when you try to represent data grouped into unequal widths. On a histogram you plot frequency density and the area of a bar on a histogram then corresponds to (is proportional to) the frequency. If you use a frequency polygon for unequal classes, you still use frequency density, but the area under the polygon does not correspond exactly to the frequency. So, to answer your question, a histogram has the advantage that the area is proportional to the frequency for unequal class widths or, more simply, it is easier to use a histogram for unequal class widths. There is another, simpler answer to your question, but it is based on a misunderstanding. If the question is "why is a bar chart used instead of a frequency polygon", you can say that a bar chart is used with un-grouped discrete data and a frequency polygon is for grouped data. Now some people, including teachers, use the word "histogram" when they should really say "bar chart" and ask why is a histogram used instead of a polygon. They expect the answer, "when the histogram is a bar chart of un-grouped data, because a polygon is for grouped data whereas a histogram can represent grouped or un-grouped data". Although, in a general sense a histogram is a type of bar chart, in the context of this question a histogram and a bar chart are not equivalent and a histogram is not used for un-grouped data. Hope some of this helps.

Question:I am currently doing the MayField coursework for my statistics GCSE. My teacher asked me to do a histogram. So far I have grouped the data (with unequal calss intervals). I have done a frequency polygon and wrote comments on it. However, my teacher mentioned about adding a histogram. The histograms are not clear with unequal classwidths. If I need to add a histogram, then I have to re-group the data so that the classwidths are equal (so the histogram is clear). My question is... Which is better, Histogram/Frequency Polygon? Do I need to add a Histogram, provided I am already giving a Frequency Polygon? Please Help. Thanks in advance.

Answers:Ive done that coursework in s10 and trust me you can still achieve full marks without doing a histogram. Just stick with the frequency polygon trust me :) x

From Youtube

Birth data: using Autograph for histograms :This video takes the set of 1175 ages of Mothers having babies and creates a histogram in Autograph. Douglas Butler discusses how Autograph copes with the tricky concepts of: discrete and continuous variables, and of frequency density, particularly when the class intervals are not equal.