Data Distribution Gathering Measures of Central Tendency Algebra
Data – Distribution & Gathering
Measures of Central Tendency Algebra 2 – 8. 1 • • Mean Median Mode Expected Value or Weighted Average – Multiply each value by its “weight” or probability – Example: Expected grade with our term weights
Weighted Average & Expected Outcomes • For numerical data, the weighted average of all of those outcomes is called the expected value for that experiment. • The probability distribution for an experiment is the function that pairs each outcome with its probability. • To find either – multiply the outcome by the number of occurrences the divide by the total number of trials
Example 1: Weighted Average A weighted average is a mean calculated by using frequencies of data values. Suppose that 30 movies are rated as follows: weighted average of stars =
Example 2: Finding Expected Value The probability distribution of successful free throws for a practice set is given below. Find the expected number of successes for one set.
Example 2 Continued Use the weighted average. Simplify. The expected number of successful free throws is 2. 05.
Data Spread • Minimum and maximum • Quartiles (First, Median, Third & Maximum) – Show in Box and Whisker Plot • Variance & Standard Deviation – Difference from the mean – How spread the data is and where are the outliers
Example – Using Calculator • In the 2003 -2004 American League Championship Series, the New York Yankees scored the following numbers of runs against the Boston Red Sox: 2, 6, 4, 2, 4, 6, 6, 10, 3, 19, 4, 4, 2, 3. Identify the outlier, if any, and describe how it affects the mean and standard deviation.
Using the TI-84 Calculator Step 1 Enter the data values into list L 1 on a graphing calculator. Step 2 Find the mean and standard deviation. On the graphing calculator, press , scroll to the CALC menu, and select 1: 1 -Var Stats. The mean is about 5. 4, and the standard deviation is about 4. 3.
Using The Calculator (Cont) Step 3 Identify the outliers. Look for the data values that are more than 3 standard deviations away from the mean in either direction. Three standard deviations is about 3(4. 3) = 12. 9. – 12. 9 +12. 9 5. 4 18. 3 Mean Values less than – 7. 5 and greater than 18. 3 are outliers, so 19 is an outlier. – 7. 5 Step 4 Remove the outlier to see the effect that it has on the mean and standard deviation.
Normal Distribution • Results from plotting data and drawing curve – Histogram, bar graph • Shape is a Bell Curve – Symmetric – No Gaps – Equal measures on both sides • No Skew • Mean and Median are Equal
Normal Distribution • Mean – μ – Average of the data • Standard Deviation – σ – Divides data into blocks – Equal units from mean (center) • Between -1σ and +1σ – Approximately 68% of results • Between -2σ and +2σ – Approximately 95% of results • Between -3σ and +3σ – Approximately 99. 7% of results
Visual Representation • Curve with Mean and Standard Deviations
Example • 500 Students with mean score of 350 and standard deviation of 25.
Standard Normal Value • Percentage of results in normally distributed data that fall within a range • Subtract low end % from high end % • Use table to determine value Example: Test scores normally distributed Mean of 75, Std Dev 8 Probability less than 87
Gathering Data (Alg 2 – 8. 2) • Vocabulary – Population – Census – Sample • Random Sample vs Biased Sample – Random – everyone has an equal CHANCE of being selected – Biased – not representative – groups under or over represented • Use samples to estimate overall populations
Gathering Data (Alg 2 – 8. 3) • Methods of data collection – Sampling – Survey • Bias • Representative – Experiment – involved, do something, compare with inputs • Treatment Group • Control Group – Observational Study – just watch or observe, take notes • When to use each
Sampling Types (Alg 2 – 8. 5) • • • Simple Random Systematic Stratified Cluster Convenience Self-Selected
Margin of Error • Interval, centered on sample percent, that accounts for possible differences or shortfalls • Shows the area where the result is likely to be • + & - the margin of error from sample result • If overlap – can’t be sure – Example: 54 % for and 46 % against with margin of error of 5 % – Because regions overlap, can’t be sure of opinion
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