Introduction and Descriptive Statistics

WHAT IS STATISTICS? STATISTICS The science of collecting, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions. Statistics is a science that helps us make better decisions in business and economics as well as in other fields. Statistics teaches us how to summarize, analyze, and draw meaningful inferences from data that then lead to improve decisions. These decisions that we make help us improve the running, for example, a department, a company, the entire economy, etc.

Using Statistics (Two Categories) Descriptive Statistics Collect Organize Summarize Display Analyze Inferential Statistics Predict and forecast value of population parameters Test hypothesis about value of population parameter based on sample statistic Make decisions

Samples and Populations A population consists of the set of all measurements for which the investigator is interested. A sample is a subset of the measurements selected from the population. A census is a complete enumeration of every item in a population.

Why Sample? Census of a population may be: Impossible Impractical Too costly

Parameter Versus Statistic PARAMETER A measurable characteristic of a population. STATISTIC A measurable characteristic of a sample.

Types of Data - Two Types Qualitative - Categorical or Nominal: Examples are- Color Gender Nationality Quantitative - Measurable or Countable: Examples are- Temperatures Salaries Number of points scored on a 100 point exam

Quantitative Variables - Classifications Quantitative variables can be classified as either discrete or continuous. A. Discrete variables: can only assume certain values and there are usually “gaps” between values. EXAMPLE: the number of bedrooms in a house, or the number of hammers sold at the local Home Depot (1,2,3,…,etc). B. Continuous variable can assume any value within a specified range. EXAMPLE: The pressure in a tire, the weight of a pork chop, or the height of students in a class.

Summary of Types of Variables 9

Scales of Measurement Nominal Scale - groups or classes Gender Ordinal Scale - order matters Ranks (top ten videos) Interval Scale - difference or distance matters – has arbitrary zero value. Temperatures (0F, 0C), Likert Scale Ratio Scale - Ratio matters – has a natural zero value. Salaries

Population and Sample Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values. The sample mean is the sum of all the sample values divided by the total number of sample values. EXAMPLE:

The Median MEDIAN The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. PROPERTIES OF THE MEDIAN There is a unique median for each data set. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. It can be computed for ratio-level, interval-level, and ordinal-level data. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. EXAMPLES: The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The heights of four basketball players, in inches, are: 76, 73, 80, 75 Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5

Quartiles, Deciles and Percentiles The standard deviation is the most widely used measure of dispersion. Alternative ways of describing spread of data include determining the location of values that divide a set of observations into equal parts. These measures include quartiles, deciles, and percentiles. To formalize the computational procedure, let Lp refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L33 and if we wanted the median, the 50th percentile, then L50. The number of observations is n, so if we want to locate the median, its position is at (n + 1)/2, or we could write this as (n + 1)(P/100), where P is the desired percentile

Percentiles - Example EXAMPLE Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s Oakland, California, office. $2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787 $2,287 $1,940 $2,311 $2,054 $2,406 $1,471 $1,460 Locate the median, the first quartile, and the third quartile for the commissions earned. Step 1: Organize the data from lowest to largest value $1,460 $1,471 $1,637 $1,721 $1,758 $1,787 $1,940 $2,038 $2,047 $2,054 $2,097 $2,205 $2,287 $2,311 $2,406 Step 2: Compute the first and third quartiles. Locate L25 and L75 using:

The Mode MODE The value of the observation that appears most frequently.

Measures of Dispersion A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from that standpoint, but it does not tell us anything about the spread of the data. For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth. A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions. RANGE Interquartile Range IQR = Difference between third and first quartile (Q3 - Q1) MEAN DEVIATION VARIANCE AND STANDARD DEVIATION There is a small difference in the formulae for population variance/standard deviation and sample variance/standard deviation.

EXAMPLE – Mean Deviation The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the mean deviation for the number of cappuccinos sold. Step 1: Compute the mean Step 2: Subtract the mean (50) from each of the observations, convert to positive if difference is negative Step 3: Sum the absolute differences found in step 2 then divide by the number of observations

Variance and Standard Deviation VARIANCE The arithmetic mean of the squared deviations from the mean. STANDARD DEVIATION The square root of the variance. The variance and standard deviations are nonnegative and are zero only if all observations are the same. For populations whose values are near the mean, the variance and standard deviation will be small. For populations whose values are dispersed from the mean, the population variance and standard deviation will be large. The variance overcomes the weakness of the range by using all the values in the population

EXAMPLE – Population Variance and Population Standard Deviation The number of traffic citations issued during the last twelve months in Beaufort County, South Carolina, is reported below: What is the population variance? Step 1: Find the mean. Step 2: Find the difference between each observation and the mean, and square that difference. Step 3: Sum all the squared differences found in step 3 Step 4: Divide the sum of the squared differences by the number of items in the population.

Sample Variance and Standard Deviation EXAMPLE The hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. What is the sample variance?

Grouping Data Dividing data into groups or classes or intervals Groups should be: Mutually exclusive Not overlapping - every observation is assigned to only one group Exhaustive Every observation is assigned to a group Equal-width (if possible) First or last group may be open-ended

Frequency Distribution Table with two columns listing: Each and every group or class or interval of values Associated frequency of each group Number of observations assigned to each group Sum of frequencies is number of observations N for population n for sample Class midpoint is the middle value of a group or class or interval Relative frequency is the percentage of total observations in each class Sum of relative frequencies = 1

Example : Frequency Distribution x f(x) f(x)/n Spending Class ($) Frequency (number of customers) Relative Frequency 0 to less than 100 30 0.163 100 to less than 200 38 0.207 200 to less than 300 50 0.272 300 to less than 400 31 0.168 400 to less than 500 22 0.120 500 to less than 600 13 0.070 184 1.000 Example of relative frequency: 30/184 = 0.163 Sum of relative frequencies = 1

Cumulative Frequency Distribution x F(x) F(x)/n Spending Class ($) Cumulative Frequency Cumulative Relative Frequency 0 to less than 100 30 0.163 100 to less than 200 68 0.370 200 to less than 300 118 0.641 300 to less than 400 149 0.810 400 to less than 500 171 0.929 500 to less than 600 184 1.000 The cumulative frequency of each group is the sum of the frequencies of that and all preceding groups.

The Arithmetic Mean and Standard Deviation of Grouped Data EXAMPLE: Determine the arithmetic mean vehicle selling price given in the frequency table below. EXAMPLE Compute the standard deviation of the vehicle selling prices in the frequency table below.

Relative Dispersion: The Coefficient of Variation Symbolically, CV (coefficient of variation) σ C V= X 100% µ

Histogram A histogram is a chart made of bars of different heights. Widths and locations of bars correspond to widths and locations of data groupings Heights of bars correspond to frequencies or relative frequencies of data groupings

Histogram Example Relative Frequency Histogram Frequency Histogram

Skewness Another characteristic of a set of data is the shape. There are four shapes commonly observed: symmetric, positively skewed, negatively skewed, bimodal. The coefficient of skewness can range from -3 up to 3. A value near -3, indicates considerable negative skewness. A value such as 1.63 indicates moderate positive skewness. A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that there is no skewness present.

The Relative Positions of the Mean, Median and the Mode 31