Business Statistics презентация

About Applied Statistics

Data Structures
Classifying the Various Types of Data Sets
Data can come to you in several different forms, and it will be useful to have a basic catalog of the different kinds of data so that you can recognize them and use appropriate techniques for each. A data set consists of observations on items, typically with the same information being recorded for each item.
A piece of information recorded for every item (eg, its cost) is called a variable. The number of variables (pieces of information) recorded for each item indicates the complexity of the data set and will guide you toward the proper kinds of analyses. Depending on whether one, two, or many variables are present, you have univariate, bivariate, or multivariate data, respectively.

Univariate Data
Univariate (one-variable) data sets have just one piece of information recorded for each item. Statistical methods are used to summarize the basic properties of this single piece of information, answering such questions as:
1. What is a typical (summary) value?
2. How diverse are these items?
3. Do any individuals or groups require special attention?

Bivariate Data
Bivariate (two-variable) data sets have exactly two pieces of information recorded for each item. In addition to summarizing each of these two variables separately (each as its own univariate data set), statistical methods would also be used to explore the relationship between the two factors being measured in the following ways:
1. Is there a simple relationship between the two?
2. How strongly are they related?
3. Can you predict one from the other? If so, with what degree of reliability?
4. Do any individuals or groups require special attention?

Multivariate Data
Multivariate (many-variable) data sets have three or more pieces of information recorded for each item. In addition to summarizing each of these variables separately (as a univariate data set), and in addition to looking at the relationship between any two variables (as a bivariate data set), statistical methods would also be used to look at the interrelationships among all the items, addressing the following questions:
1. Is there a simple relationship among them?
2. How strongly are they related?
3. Can you predict one (a “special variable”) from the others? With what degree of reliability?
4. Do any individuals or groups require special attention?

QUANTITATIVE DATA: NUMBERS
Meaningful numbers are numbers that directly represent the measured or observed amount of some characteristic or quality of the elementary units, as the result of an observation of a variable. Meaningful numbers include, for example, dollar amounts, counts, sizes, numbers of employees.
If the data for a variable comes to you as meaningful numbers, then you have quantitative data (ie, they represent quantities). With quantitative data, you can do all of the usual number-crunching tasks, such as finding the average and measuring the variability.
It is straightforward to compute directly with numerical data.
There are two kinds of quantitative data, discrete and continuous, depending on the values potentially observable.

Discrete data is a count that can't be made more precise. Typically it involves integers. For instance, the number of children (or adults, or pets) in your family is discrete data, because you are counting whole, indivisible entities: you can't have 2.5 kids, or 1.3 pets.
Continuous data, on the other hand, could be divided and reduced to finer and finer levels. For example, you can measure the height of your kids at progressively more precise scales—meters, centimeters, millimeters, and beyond—so height is continuous data.

QUALITATIVE DATA: CATEGORIES
Qualitative data is defined as the data that approximates and characterizes.
This data type is non-numerical in nature. This type of data is collected through methods of observations, one-to-one interview, conducting focus groups and similar methods.
Qualitative data in statistics is also known as categorical data.
There are two kinds of qualitative data: ordinal (for which there is a meaningful ordering but no meaningful numerical assignment) and nominal (for which there is no meaningful order).

Charts, Histograms, Graphing

USING A HISTOGRAM TO DISPLAY
THE FREQUENCIES
The histogram displays the frequencies as a bar chart rising above the number line, indicating how often the various values occur in the data set. The horizontal axis represents the measurements of the data set (eg, in dollars, number of people, miles per gallon, etc.), and the vertical axis represents how often these values occur. An especially high bar indicates that many data values were found at this position on the horizontal number line, while a shorter bar indicates a less common value.
A histogram is a bar chart of the frequencies, not of the data.

Creating Frequency Distributions and Histograms in EXCEL

Consider the interest rate for 25-year fixed-rate home mortgages charged by mortgage companies in Seattle

NORMAL DISTRIBUTIONS
A normal distribution is an idealized, smooth, bell-shaped histogram with all of the randomness removed. It represents an ideal data set that has lots of numbers concentrated in the middle of the range, with the remaining numbers trailing off symmetrically on both sides. This degree of smoothness is not attainable by real data.

There are actually many different normal distributions, all symmetrically bell-shaped. They differ in that the center can be anywhere, and the scale (the width of the bell) can have any size.

BIMODAL DISTRIBUTIONS WITH
TWO GROUPS
It is important to be able to recognize when a data set consists of two or more distinct groups so that they may be analyzed separately, if appropriate. This can be seen in a histogram as a distinct gap between two cohesive groups of bars. When two clearly separate groups are visible in a histogram, you have a bimodal distribution. Literally, a bimodal distribution has two modes, or two distinct clusters of data.

Task 1. Using the table below build different types of charts and graphing

TASK 2 Consider the yields (as an interest rate, in percent per year) of municipal bonds, as shown in Table 3.8.1.
a. Construct a histogram of this data set.
b. Describe the shape of the distribution.

Landmark Summaries
Interpreting Typical Values and Percentiles
WHAT IS THE MOST TYPICAL VALUE?
There are three different ways to obtain such a summary measure:
1. The average or mean, which can be computed only for meaningful numbers (quantitative data).
2. The median, or halfway point, which can be computed either for ordered categories (ordinal data) or for numbers.
3. The mode, or most common category, which can be computed for unordered categories (nominal data), ordered categories, or numbers.

The Average: A Typical Value for
Quantitative Data
The average (also called the mean) is the most common method for finding a typical value for a list of numbers, found by adding up all the values and then dividing by the number of items. The sample average expressed as a formula is:

Excel’s Average function

The Weighted Average: Adjusting
for Importance
The weighted average is like the average, except that it allows you to give a different importance, or “weight,” to each data item. The weighted average gives you the flexibility to define your own system of importance when it is not appropriate to treat each item equally.

The Median: A Typical Value for
Quantitative and Ordinal Data
The median is the middle value; half of the items in the set are larger and half are smaller. Thus, itmust be in the center of the data and provide an effective summary of the list of data. You find it by first putting the data in order and then locating the middle value. To be precise, you have to pay attention to some details; for example, you might have to average the two middle values if there is no single value in the middle.
One way to define the median is in terms of ranks. Ranks associate the numbers 1, 2, 3,…,n with the data values so that the smallest has rank 1, the next smallest has rank 2, and so forth up to the largest, which has rank n.

The Mode: A Typical Value Even
for Nominal Data
The mode is the most common category, the one listed most often in the data set. It is the only summary measure available for nominal qualitative data because unordered categories cannot be summed (as for the average) and cannot be ranked (as for the median). The mode is easily found for ordinal data by ignoring the ordering of the categories and proceeding as if you had a nominal data set with unordered categories.
Mode is the value which occurs most frequently. The mode may not exist, and even if it does, it may not be unique.

Example 1
The moda of the ungrouped data: 20, 18, 15, 15, 14, 12, 11, 9, 7, 6, 4, 1 is 15
Example 2
{2, 2, 2, 4, 5, 6, 7, 7, 7}
Mode = 2 or 7 (Bimodal)

Task
To calculate:
The Average and The Weighted Average and Rang
1) 24, 26, 26, 29, 33, 36, 37
2) 24, 26, 26, 29, 33, 36, 37, 45
Median:
1) 37, 26, 29, 33, 24, 36, 26
2) 45, 26, 36, 29, 33, 37, 24, 26,
Moda
1) 24, 26, 26, 29, 33, 36, 37
2) 36, 26, 26, 29, 33, 36, 37, 45

Consider the profits of health care companies in the Fortune 500, as shown in Table 4.3.3
a. Draw a histogram of this data set, and briefly describe the shape of the distribution.
b. Find the profit of the average firm.
c. Find the median profit level.
d. Compare the average and median; in particular, which is larger? Is this what you would expect for a distribution with this shape?

Task
Consider the loan fees charged for granting home mortgages,
as shown in Table 4.3.8, for Dallas TX, 30-year fixed rate for home purchase, with credit Score 740+, and with 20% down payment. These are given as a percentage of the loan amount and are one-time fees paid when the loan is closed.
a. Find the average loan fee.
b. Find the median loan fee.
c. Find the mode.
d. Which summary is most useful as a description of the “typical” loan fee, the average, median, or mode? Why?

Probability
Understanding Random Situations
Probability is a measure quantifying the likelihood that events will occur.
Probability. How likely something is to happen. Many events can't be predicted with total certainty.
If events are mutually incompatible (they can’t occure at the same time) and they constitute all possibilities for a given situation, the sum of their individual probabilities is one. For example, a fair cubical die has a probability
pi = 1/6
of landing with the top face showing one dot (i=1) facing up; indeed pi = 1/6 for all i. The probability that a fair die will land showing a top face of either a 1 or a 2 or a 3 or a 4 or a 5 or a 6 is
Σpi = 1

A probability tree is a picture indicating probabilities and conditional probabilities for combinations of two or more events. We will begin with an example of a completed tree and follow up with the details of how to construct the tree.
Probability trees are closely related to decision trees, which are used in finance and other fields in business.