STATISTICAL REPRESENTATION

CUMULATIVE FREQUENCY CURVE

Cumulative Frequency Graph , also known as an Ogive, is a curve showing the cumulative frequency for a given set of data. The cumulative frequency is plotted on the y-axis against the data which is on the x-axis for UN-grouped data.

Cumulative Frequency Graph If we are given a table containing continuous data, we can find a running total of the frequency. This is called Cumulative Frequency. Height (m) Frequency Cumulative Frequency In context 7 7 7 people less than 1.4m 8 15 15 people less than 1.6m 22 37 37 people less than 1.8m 3 40 40 people less than 2.0m

We can therefore plot a graph: ·       The x-axis (horizontal) will be Height(m) ·       The y-axis will (vertical) will be Cumulative Frequency ·       We plot the numbers in red ·       The graph starts at 1.2m, because this was the lowest value

We can now use this graph to estimate a number of key values In this case n=40 (the total number of people Median: On a cumulative frequency graph we find  value We can read this off the graph to get 1.64m Lower Quartile: On a cumulative frequency graph we find  value We can read this off the graph to get 1.48m Upper Quartile: On a cumulative frequency graph we find  value We can read this off the graph to get 1.73m From this we can deduce the IQR = UQ - LQ = 1.73m-1.48m=0.25m

Histogram – the A/A* graph! Suppose you are given a table of continuous data (see below). Given a class eg.  the class width is  In the table below, you can see how the class widths change. In this case, we will need to construct a histogram to represent the data. In a histogram, the AREA of the bars equals the FREQUENCY To achieve this, we need to calculate a value called the FREQUENCY DENSITY Height (m) Frequency class-width Frequency density 5 0.2 25 12 0.3 40 15 0.3 50 2 0.1 20 We now draw a graph with: ·       Height(m) on the x-axis ·       Frequency density on the y-axis

The next section considers how to read graphs to find an average

Finding the mean and median from a frequency graph A frequency graph to show the frequency of scores in a test                       This graph can be turned into a frequency table Mark Midpoint Frequency 5 8 15 12 25 11 35 3 TOTAL 34

Frequency diagrams and polygons

Bar charts and frequency diagrams. Pie charts are useful for showing proportions, but different types of chart have to be used for representing other kinds of data. A number of these charts are described in this section..This frequency diagram shows the heights of 200 people:

You can construct a frequency polygon by joining the midpoints of the tops of the bars.
Frequency polygons are particularly useful for comparing different sets of data on the same diagram.

Constructing a frequency polygon

Midpoints are marked on each bar and joined together

Frequency diagrams and polygons

This frequency diagram shows the heights of 200 people:

You can construct a frequency polygon by joining the midpoints of the tops of the bars.
Frequency polygons are particularly useful for comparing different sets of data on the same diagram.

Constructing a frequency polygon

STEM AND LEAF PLOTS

Stem-and-leaf plots are a method for showing the frequency with which certain classes of values occur. You could make a frequency distribution table or a histogram for the values, or you can use a stem-and-leaf plot and let the numbers themselves to show pretty much the same information.
For instance, suppose you have the following list of values: 12, 13, 21, 27, 33, 34, 35, 37, 40, 40, 41. You could make a frequency distribution table showing how many tens, twenties, thirties, and forties you have:

Frequency Class	Frequency
10 - 19	2
20 - 29	2
30 - 39	4
40 - 49	3

You could make a histogram, which is a bar-graph showing the number of occurrences, with the classes being numbers in the tens, twenties, thirties, and forties:

(The shading of the bars in a histogram isn't necessary, but it can be helpful by making the bars easier to see, especially if you can't use color to differentiate the bars.)
The downside of frequency distribution tables and histograms is that, while the frequency of each class is easy to see, the original data points have been lost. You can tell, for instance, that there must have been three listed values that were in the forties, but there is no way to tell from the table or from the histogram what those values might have been.

On the other hand, you could make a stem-and-leaf plot for the same data:

The "stem" is the left-hand column which contains the tens digits. The "leaves" are the lists in the right-hand column, showing all the ones digits for each of the tens, twenties, thirties, and forties. As you can see, the original values can still be determined; you can tell, from that bottom leaf, that the three values in the forties were 40, 40, and 41.
Note that the horizontal leaves in the stem-and-leaf plot correspond to the vertical bars in the histogram, and the leaves have lengths that equal the numbers in the frequency table.

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That's pretty much all there is to a stem-and-leaf plot. You're just listing out how many entries you have in certain classes of numbers, and what those entries are. Here are some more examples of stem-and-leaf plots, containing a few additional details.

• Complete a stem-and-leaf plot for the following list of grades on a recent test:
73,  42,  67,  78,  99,  84,  91,  82,  86,  94

I'll use the tens digits as the stem values and the ones digits as the leaves. For convenience sake, I'll order the list, but this is not required:

42,  67,  73,  78,  82,  84,  86,  91,  94,  99
Since I know where these data points came from ("a recent test"), I'll use a title. Then my plot looks like this: Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved

The above is the simplest case for stem-and-leaf plots, but even the "complicated" cases aren't much more complex. 


Stem-and-Leaf Plots: Examples

• Subjects in a psychological study were timed while completing a certain task. Complete a stem-and-leaf plot for the following list of times:

7.6,  8.1,  9.2,  6.8,  5.9,  6.2,  6.1,  
5.8,  7.3,  8.1,  8.8,  7.4,  7.7,  8.2

First, I'll reorder this list:

5.8,  5.9,  6.1,  6.2,  6.8,
  7.3,  7.4,  7.6,  7.7,  8.1,
   8.1,  8.2,  8.8,  9.2
These values have one decimal place, but the stem-and-leaf plot makes no accomodation for this. The stem-and-leaf plot only looks at the last digit (for the leaves) and all the digits before (for the stem). So I'll have to put a "key" or legend on this plot to show what I mean by the numbers in this plot. The ones digits will be the stem values, and the tenths will be the leaves.

Properly, every stem-and-leaf plot should have a key.

• Complete a stem-and-leaf plot for the following two lists of class sizes:

Economics 101:  9,  13,  14,  15,  16,  16,  17,  19,  20,  21,  21,  22,  25,  25,  26
Libertarianism:  14,  16,  17,  18,  18,  20,  20,  24,  29
This example has two lists of values. Since the values are similar, I can plot them all on one stem-and-leaf plot by drawing leaves on either side of the stem. I will use the tens digits as the stem values, and the ones digits as the leaves. Since "9" (in the Econ 101 list) has no tens digit, the stem value will be "0". Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved

• Complete a stem-and-leaf plot for the following list of values:
100,  110,  120,  130,  130,  150,  160,  170,  170,  190,
210,  230,  240,  260,  270,  270,  280.  290,  290
Since all the ones digits are zeroes, I'll do this plot with the hundreds digits being the stem values and the tens digits being the leaves. I can do the plot like this:

...but the leaves are fairly long this way, because the values are so close together. To spread the values out a bit, I can break each leaf into two. For instance, the leaf for the two-hundreds class can be split into two classes, being the numbers between 200 and 240 and the numbers between 250 and 290. I can also reverse the order, so the smaller values are at the bottom of the "stem". The new plot looks like this:

For very compact data points, you can even split the leaves into five classes, like this:

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• Complete a stem-and-leaf plot for the following list of values:

23.25,  24.13,  24.76,  24.81,  24.98,  25.31,  25.57,  25.89,  26.28,  26.34,  27.09
If I try to use the last digit, the hundredths digit, for these numbers, the stem-and-leaf plot will be enormously long, because these values are so spread out. (With the numbers' first three digits ranging from 232 to 270, I'd have thirty-nine leaves, most of which would be empty.) So instead of working with the given numbers, I'll round each of the numbers to the nearest tenth, and then use those new values for my plot. Rounding gives me the following list:

23.3,  24.1,  24.8,  24.8,  25.0,  25.3,  25.6,  25.9,  26.3,  26.3,  27.1

Then my plot looks like this:


  

Naturally, when you're drawing a stem-and-leaf plot, you should use a ruler to construct a neat table, and you should label everything clearly.

linear programming

Linear Programming: Introduction

Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the "best" production levels for maximal profits under those conditions.
In "real life", linear programming is part of a very important area of mathematics called "optimization techniques". This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more. In algebra, though, you'll only work with the simple (and graphable) two-variable linear case.
The general process for solving linear-programming exercises is to graph the inequalities (called the "constraints") to form a walled-off area on the x,y-plane (called the "feasibility region"). Then you figure out the coordinates of the corners of this feasibility region (that is, you find the intersection points of the various pairs of lines), and test these corner points in the formula (called the "optimization equation") for which you're trying to find the highest or lowest value.

• Find the maximal and minimal value of z = 3x + 4y subject to the following constraints:
The three inequalities in the curly braces are the constraints. The area of the plane that they mark off will be the feasibility region. The formula "z = 3x + 4y" is the optimization equation. I need to find the (x, y) corner points of the feasibility region that return the largest and smallest values of z.
My first step is to solve each inequality for the more-easily graphed equivalent forms:

It's easy to graph the system:   Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved

To find the corner points -- which aren't always clear from the graph -- I'll pair the lines (thus forming a system of linear equations) and solve:


y = –( 1/2 )x + 7y = 3x	y = –( 1/2 )x + 7y = x – 2	y = 3x y = x – 2
–( 1/2 )x + 7 = 3x–x + 14 = 6x14 = 7x 2 = x y = 3(2) = 6	–( 1/2 )x + 7 = x – 2 –x + 14 = 2x – 4 18 = 3x 6 = x y = (6) – 2 = 4	3x = x – 2 2x = –2x = –1 y = 3(–1) = –3
corner point at (2, 6)	corner point at (6, 4)	corner pt. at (–1, –3)

So the corner points are (2, 6), (6, 4), and (–1, –3).
Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. So, to find the solution to this exercise, I only need to plug these three points into "z = 3x + 4y".

(2, 6):      z = 3(2)   + 4(6)   =   6 + 24 =   30
(6, 4):      z = 3(6)   + 4(4)   = 18 + 16 =   34
(–1, –3):  z = 3(–1) + 4(–3) = –3 – 12 = –15
Then the maximum of z = 34 occurs at (6, 4),
and the minimum of z = –15 occurs at (–1, –3).

HERE A VIDEO EXPLANATION: