Statistics Descriptive Statistics

Statistics Introduction Descriptive Statistics Collections, organizations, summary and presentation of data Inferential Statistics Determining relationships between variables Testing hypotheses Making predictions Using distributions to make predictions

Descriptive Statistics Examples: U.S. Census, surveys

Today’s Goals Collect a sample Compare sampling techniques Central Tendency Mean, Median Quartiles Spread Standard Deviation Range, Inter-quartile range

Sampling Sampling Techniques

Convenience, Biased Simple Random Systematic The group of 100 rectangles represents the population. You are going to take samples of 10 to attempt to describe the traits of the population.

Statistical Measures Central Tendency Mode, Mean, Median Using Your Calculator

Central Tendency Mode Mean Median

Mean Numeric data is added and the result is divided by the number of items of data “Average” (a misleading nickname) Formulas on the right give a method for calculating mean from: A full set of data A frequency table or histogram

Two methods for calculating Mean If you have all the data: Add values and divide by n If you have a frequency table Calculate all products of value times frequency Add those products Divide by the sum of the frequencies

Median Order all data by size Select the “middle” value If there is an even number of items, take the mean of the middle two

Quartiles Once you have determined the median, use that center to divide the rest of the data into two groups. The median of the lower group is called the 1 st (or lower) quartile. Notation: Q1. The median of the upper group is called the 3 rd (or upper) quartile. Notation: Q3.

Measures of Spread Variance and Standard Deviation

Central Tendency…Limitations Describe the data presented in the Excel sheet in words

Measures of Spread Inter-quartile Range (IQR) If it is more appropriate to use the median to describe the center of the date… …then it is more appropriate to use range and inter-quartile range to describe the spread. IQR = Q3 – Q1 Range = Max – Min

Measures of Spread Variance For each data item Find the distance that an item “deviates from the mean” Square the “deviation” value Multiply this by the frequency f Finally, Find the sum of each of these calculations and divide by the sum of the frequencies

Standard Deviation by Hand The following is a set of IB Scores 7, 4, 6, 3, 7, 3 Calculate the Mean, Variance and Standard Deviation by hand

Variance We will not use variance much on its own, but it is a valuable measure of spread

Standard Deviation The square root of variance This is the most widely used measure of spread

Standard Deviation Sample, or “unbiased” Notation varies Population

Calculators and Spreadsheets to the Rescue!!!!!! Entering the Table: STAT, EDIT Bringing up lists on the home screen: the list names (L1, L2, …) are above the number keys.

Calculators and Spreadsheets to the Rescue!!!!!! One variable statistical measures STAT, CALC, 1-Var Stats List name The same, but from a frequency table STAT, CALC, 1-Var Stats Data List, Frequency List Sorting STAT, SortA[scending] and SortD[escending]

Calculators and Spreadsheets to the Rescue!!!!!! Common Error Messages: SYNTAX: you typed something incorrectly INVALID DIM: some function is comparing lists whose sizes don’t match. Either fix your lists by entering the STAT menu and selecting EDIT, or turn off the statistical plots (Plot1, Plot2 or Plot3).

Population vs. Sample S (our alphabet): this is the sample standard deviation. If your data is a sample from the population, use this. Sigma (greek letter): this is the population standard deviation. If your data represents an entire population of something, use this.

Activity (Option 1) Web site: ap.cfm ap.cfm For your assigned continent, make a table of “Human Development Indices” Calculate: Central Tendency Variance Standard Deviation

Activity (Option 2) Construct a table of values for “Ratio of Forearm to Hand” Calculate: Central Tendency Variance Standard Deviation You may use your calculator

Homework Central Tendency (13.3) 4, 5 Standard Dev. (13.4) 2 Quartiles, IQR (13.5) Raw data: 1 (a, d) Grouped data: 2a

Displaying Data Data in Tables: Frequency Tables, Stem-and-Leaf Graphs: Histograms, Box Plots Cumulative Frequency Curves

Frequency Tables The first column can contain either: Values for data (“x”) Intervals of pre- determined width It is important to understand how rounding (or the lack of it) occurs to assign data to the correct intervals IntervalFreq.Tally

Stem-and-Leaf The “Stem” is the numerical data, truncated in front of the last digit. The “Leaf” is the last digit Purpose Turn on its side for a frequency bar graph Original data can be recreated StemLeaf

Frequency Histograms This is a bar graph where adjacent bars touch one another. For now, we will work with intervals or equal width (though histograms can be made with intervals of unequal widths) Horizontal axis: categories Vertical axis: frequency

Box Plots One-dimensional display of the Minimum, Q1, Median, Q3 and Maximum Calculator: use the Stat Plot The box plot is the 5 th option

Displaying Data Calculators Cumulative Frequency Percentiles

Start of Class 1) Clear all lists through “Clrlist L1,L2,…” 2) Enter raw “Poverty” data into L1 3) Enter your frequency table with class intervals of width TWO into L2 and L3. Use the midpoint of the interval as your value.

Cumulative Frequency Table (page 482) Use the table for Customers and Frequency How is the cumulative frequency calculated?

Cumulative Frequency or “Ogive” (page 482) Horizontal axis: top end of the category for # of customers Vertical axis: cumulative frequency Cumulative frequency curve Plot points (“Top end of category”, ”cumulative frequency”) This curve allows us to estimate: Median, Quartiles, Percentiles, Inter-quartile Range

Cumulative Frequency Curve Trends in the curve Increasing slope to decreasing slope Concave UP to concave DOWN Finding the Median and Quartiles Finding Inter-quartile Range How do these measures compare to the measures obtained from using the midpoints of each category?

Activity (subject to change) Page 489, #10 (by hand, and on graph paper!) Although the text does not require it, please calculate the median and inter-quartile range as well After doing this by hand, enter the frequency table in your calculator and check your work

Descriptive Statistics Test (Chapter 13) next class Measures of central tendency mean, median, mode from raw data and from frequency tables *Quartiles and percentiles Measures of spread standard deviation, range, inter-quartile range Tables and graphs Box plots, frequency histograms, cumulative frequency tables and curves Interpreting and creating your own