DO NOW1/29/14 Use the gas price data to answer the following questions. {$4.79, $4.60, $4.75, $4.66, $4.60} 1.Find the mean of the data (hint: average) 2.Find the median of the data (hint: middle) 3.Find the mode of the data (hint: most) 4.Which of these do you think best describes the center of the data? Why?

DO NOW1/30/14 1.Which of the distributions below do you think best represents the distribution of income in the United States? Why? $ $$$$$ Yearly Income $ $$$$$ Yearly Income $ $$$$$ Yearly Income FrequencyFrequency FrequencyFrequency FrequencyFrequency

Unit 1: Income Topic: Probability Distributions ChiArts Financial Statistics

Probability Distributions Why do we care? – We want to describe the center (average, typical case) of the data. – We want to describe a data point in relation to others. Our motivation: – What is the average American’s income? – What is the average income for a specific career? These are two very different questions!

Measures of Center Mean – Average, or the sum of the data divided by the total number of values in the data set Median – Middle number (when numbers are in ORDER!) Mode – Most frequent number (may be more than one!)

Example #1 – Gas prices Premium gas price data: $4.79, $4.60, $4.75, $4.66, $4.60 Mean = Median: $4.60, $4.60, $4.66, $4.75, $4.79 Mode: $4.60, $4.60, $4.66, $4.75, $4.79 Which of these best describes the center? $

Example #2 - Age But the mean doesn’t always best describe the center! Cousin Age data: 17, 17, 18, 17, 17, 1, 1, 1 Mean = Median: 1, 1, 1, 17, 17, 17, 17, 18 Mode: 1, 1, 1, 17, 17, 17, 17, 18 Which of these best describes the typical case? Median = (17+17)/2 = 17

Graphical Representations Histogram: a bar graph of a frequency distribution that organizes data into equal ranges. Data range “bins” (here, bin width = 5) Frequency

Graphical Representations Special types of distributions Skewed Right (Positive skew) Skewed Left (Negative skew) Normal (Bell Curve)

Graphical Representations Measures of center Normal Distribution (Bell Curve) Mean = Median = Mode

Graphical Representations Measures of center Skewed Right (Positive skew) Mode Median Mean

Graphical Representations Measures of center Skewed Left (Negative skew) Mode Median Mean

Normal Distribution (Bell Curve) 50 th percentile 50% of the data lie at or to the left of the value at this line 50% Percentiles Percentile: a measure that tells us what percent of the total frequency is at or below that measure. 84 th percentile 84% of the data lie at or to the left of the value at this line 84%

Example #3 - Height You can calculate percentiles for a data set Ordered height data (inches) Percentile = data point in position P, where P= p(n+1) p – percentile of interest n – number of data points

Example #3 - Height Ordered height data (inches) Calculate the 70 th percentile for the data above. 70 th Percentile = data point in position P, where P= 70(49+1) = Position 1 Position th Percentile = 69 inches This means 70% of these people are 69 inches or shorter.

Example #3 – Height Data in Histogram form 70 th Percentile = 69 inches 70% of the data are at or to the left of this line

Probability Distributions Now you try! – Station 1: Vocabulary Graphic Organizer – Station 2: Practice with mean, median & mode – Station 3: Practice with percentiles – Station 4: Analyze data* *challenge station Assignment #1.3: Redo Unit 1 pretest 1. Remember, it is due Thursday!