Introduction to Statistics Chapter 1 Introduction to Statistics

Data collections of observations (such as measurements, genders, survey responses)

Statistics It is the study of the • collection, • organization, • analysis, • interpretation and • presentation of data

Population, Sample and Census The collection of all individuals or items under consideration in a statistical study. Sample That part of the population from which information is obtained. Census Collection of data from every member of a population.

Figure 1.1 Relationship between population and sample Insert Figure 1.1 Relationship between population and sample

Parameter

Statistic

Simple Random Sampling; Simple Random Sample Simple random sampling: A sampling procedure for which each possible sample of a given size is equally likely to be the one obtained. Simple random sample: A sample obtained by simple random sampling. There are two types of simple random sampling. One is simple random sampling with replacement, whereby a member of the population can be selected more than once; the other is simple random sampling without replacement, whereby a member of the population can be selected at most once.

Basic Data Types Quantitative ( or numerical or measurement ) data Categorical (or qualitative or attribute) data

Quantitative Data

Categorical Data

Working with Quantitative Data Quantitative data can further be described by distinguishing between discrete and continuous types.

Discrete Data Discrete data result when the number of possible values is either a finite number or a ‘countable’ number (i.e. the number of possible values is 0, 1, 2, 3, . . .) Example: The number of eggs that a hen lays, Test score, shoe size, age, world ranking, number of brothers etc. The number of eggs that a hen lays is discrete quantitative measure because it is numeric but can only be a whole number

Continuous (numerical) data Continuous Data Continuous (numerical) data result from infinitely many possible values that correspond to some continuous scale that covers a range of values without gaps, interruptions, or jumps Example: Height, weight, length, amounts of milk from cows etc. Height is continuous quantitative measure because it can take any numerical value in a particular range. The amount of milk that a cow produces; e.g. 2.343115 gallons per day.

Decide whether the following data are qualitative, discrete quantitative or continuous quantitative. 1. Number of cars 2. Mass of an object 3. distance of FAU from home 4. Day of the week 5. Color of cars 6. Pocket money 7. Favorite soccer team 8. World ranking 9. Birth place 10. Age

Classification of Data using levels of measurement Nominal level of measurement Ordinal level of measurement Interval level of measurement Ratio level of measurement

Nominal Level Nominal level of measurement is characterized by data that consist of names, labels, or categories only, and the data cannot be arranged in an ordering scheme (such as low to high) Examples: Survey responses yes, no, undecided Political Party: The political party affiliation of survey respondents (Democrat, Republican, Independent, other)

Ordinal Level Ordinal level of measurement involves data that can be arranged in some order, but differences (obtained by subtraction) between data values either cannot be determined or are meaningless Example: Course grades A, B, C, D, or F Universities rank in USA (like 1st, 2nd, 3rd, 4th,…)

Interval Level Interval level of measurement is like the ordinal level, with the additional property that the difference between any two data values is meaningful. However, data at this level do not have a natural zero starting point (where none of the quantity is present). Example: Body temperatures of 96.2 F and 98.6 F (There is no natural starting point. The value of 0 F might seem like a starting point, but it is arbitrary and does not represent the total absence of heat.) Years: 1000, 2000, 1776, and 1492. (Time did not begin in the year 0, so the year 0 is arbitrary instead of being a natural zero starting point representing “no time.”)

Ratio Level Ratio level of measurement Is the interval level with the additional property that there is also a natural zero starting point (where zero indicates that none of the quantity is present); for values at this level, differences and ratios are meaningful. Example: Prices: Prices of college textbooks ($0 represents no cost, a $100 book costs twice as much as a $50 book.) Distances: Distances (in miles) travelled by cars (0 mile represents no distance travelled, and 60 miles is twice as far as 30 miles)

Summary - Levels of Measurement Nominal - categories only Ordinal - categories with some order Interval - differences but no natural starting point Ratio - differences and a natural starting point

Summarizing and Graphing Data Chapter 2 Summarizing and Graphing Data

Important Characteristics of Data 1. Center: A representative or average value that indicates where the middle of the data set is located. 2. Variation: A measure of the amount that the data values vary. 3. Distribution: The nature or shape of the spread of data over the range of values (such as bell-shaped, uniform, or skewed). 4. Outliers: Sample values that lie very far away from the vast majority of other sample values. 5. Time: Changing characteristics of the data over time.

Frequency Distribution (or Frequency Table) In statistics, a frequency distribution is an arrangement 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.

Pulse Rates of Females and Males

Frequency Distribution Pulse Rates of Females The frequency for a particular class is the number of original values that fall into that class.

Lower Class Limits Lower Class Limits The Lower class limits are the smallest numbers that can actually belong to different classes. Lower Class Limits

Upper Class Limits Upper Class Limits The upper class limits are the largest numbers that can actually belong to different classes. Upper Class Limits

Class Boundaries Class Boundaries The class boundaries are the numbers used to separate classes, but without the gaps created by class limits. 59.5 69.5 79.5 89.5 99.5 109.5 119.5 129.5 Class Boundaries

Class Midpoints ooooc Class Midpoints 64.5 74.5 84.5 94.5 104.5 114.5 124.5 Class Midpoints

Class Width Class Width Class width is the difference between two consecutive lower class limits or two consecutive lower class boundaries. Class Width 10

Constructing A Frequency Distribution 1. Determine the number of classes (should be between 5 and 20). 2. Calculate the class width (round up). class width (maximum value) – (minimum value) number of classes 3. Starting point: Choose the minimum data value or a convenient value below it as the first lower class limit. Using the first lower class limit and class width, proceed to list the other lower class limits. 5. List the lower class limits in a vertical column and proceed to enter the upper class limits. 6. Take each individual data value and put a tally mark in the appropriate class. Add the tally marks to get the frequency.

Relative Frequency Distribution . includes the same class limits as a frequency distribution, but the frequency of a class is replaced with a relative frequencies (a proportion) or a percentage frequency ( a percent) relative frequency = class frequency sum of all frequencies percentage frequency class frequency sum of all frequencies  100% =

Relative Frequency Distribution * Total Frequency = 40 * 12/40  100 = 30%

Cumulative Frequency Distribution Cumulative Frequencies

Frequency Tables

Characteristic of Normal Distribution It has a “bell” shape. The frequencies start low, then increase to one or two high frequencies, then decrease to a low frequency. The distribution is approximately symmetric, with frequencies preceding the maximum being roughly a mirror image of those that follow the maximum.

Histogram A graph consisting of bars of equal width drawn adjacent to each other (without gaps). The horizontal scale represents the classes of quantitative data values and the vertical scale represents the frequencies. The heights of the bars correspond to the frequency values.

Histogram Basically a graphic version of a frequency distribution.

Histogram The bars on the horizontal scale are labeled with one of the following: Class boundaries Class midpoints Lower class limits (introduces a small error) Horizontal Scale for Histogram: Use class boundaries or class midpoints. Vertical Scale for Histogram: Use the class frequencies.

Relative Frequency Histogram It has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies instead of actual frequencies.

Interpreting Histograms When graphed, a normal distribution has a “bell” shape. Characteristic of the bell shape are (1) The frequencies increase to a maximum, and then decrease, and (2) symmetry, with the left half of the graph roughly a mirror image of the right half. The histogram on the next slide illustrates this.

Histogram

Frequency Polygon Uses line segments connected to points directly above class midpoint values.

Relative Frequency Polygon Uses relative frequencies (proportions or percentages) for the vertical scale.

Ogive A line graph that depicts cumulative frequencies

Dot Plot Consists of a graph in which each data value is plotted as a point (or dot) along a scale of values. Dots representing equal values are stacked.

Bar Graph Uses bars of equal width to show frequencies of categories of qualitative data. Vertical scale represents frequencies or relative frequencies. Horizontal scale identifies the different categories of qualitative data. A multiple bar graph has two or more sets of bars, and is used to compare two or more data sets.

Multiple Bar Graph

Pareto Chart A bar graph for qualitative data, with the bars arranged in descending order according to frequencies

Pie Chart A graph depicting qualitative data as slices of a circle, size of slice is proportional to frequency count

Scatter Plot (or Scatter Diagram) A plot of paired (x,y) data with a horizontal x-axis and a vertical y-axis. Used to determine whether there is a relationship between the two variables.

Time-Series Graph Data that have been collected at different points in time: time-series data.