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Unit 7: Descriptive Statistics Lesson 2: Summarizing Data Objectives: 1. To use a graphing calculator to find measures of central tendency. 2. To determine which measure of central tendency is appropriate for a data set. Measures of central tendency In addition to graphing a data set, a measure of its center or average also helps to describe the data. The two most common measures of center are the mean and median. Mean: sum of data values number of data values x + x + ... + xn = 1 2 n x= Raw Data n = åx i=1 i n sum (each data value ´ frequency) x= number of data values Frequency Table with Individual Data Values n = i=1 i i n n = x= Frequency Table with Grouped Data å x × f (x ) å xi × f (xi ) i=1 åf xi = data value f (xi ) = frequency of xi mi = midpoint of interval å is short for "add them all up" sum (each midpoint ´ frequency) number of data values n = å m × f (x ) i=1 i i n n å m × f (x ) = åf i i i=1 Median: Firstly arrange the data in order. The median is bottom or top. n +1 observations from the 2 If there is an odd number of measurements, the median = middle value. If there is an even number of measurements, the median = average of the two middle values. Mode: The most common data value. This is the only measure of central tendency that can be found for qualitative data. However, it is rarely used for quantitative data. Doing these calculations by hand can be time consuming. Thankfully the TI-84 calculator can do these for you, and in this unit you may always use a calculator to find any summary statistics. How to find summary statistics on the TI-84 calculator Enter your data into a list. There are two ways to do this: using a single list, or using a data list and a frequency list. Option 1 (L1 contains data values) Option 2 (L2 contains frequencies) To calculate the summary statistics, press STAT, scroll over to CALC, and choose option 1:1-Var Stats Option 1 Option 2 List:L1 FreqList: (leave blank) List:L1 FreqList:L2 Interpreting the output for one variable statistics* Symbol Meaning mean x x sum of all data values squares each data value, then x2 adds them up sx sample standard deviation x population standard deviation n MinX Q1 Med Q3 MaxX number of data values minimum (smallest) data value value of first (lower) quartile median value of third (upper) quartile maximum (largest) data value Note Don’t use this standard deviation (you will learn about this in AP Statistics) In Algebra 2 Trig we always use this one for the standard deviation. These five values are known collectively as the five-number summary. * We will learn about most of these statistics over the next few classes. 1. Find the mean and median of the number of siblings each student has (Example 2 from Lesson 1 notes). 2. Find the mean and median of the hand widths you measured last class (Example 3 from Lesson 1 notes). 3. Give an example of a set of five positive numbers whose median is 10 and whose mean is larger than 10. 4. In a small town of 1000 residents everyone earns $80,000 per year. a. What are the median and mean incomes for the town? 5. b. Donald Trump likes the town as it reminds him of his childhood, so he decides to move in. He earns $1,000,000,000 per year. What are the new median and mean incomes? c. Describe how Donald Trump’s earnings impact the mean and median incomes. The brightness of celestial bodies depends on many factors, two of the most important being the distance from Earth and size. The eight brightest objects in the night sky are listed below with their approximate distance from Earth (in light years). Celestial Body Distance from Earth (light years) Moon 0.000000038 Venus 0.0000048 Jupiter 0.000067 Mars 0.0000076 Mercury 0.0000095 Syrius 8.6 Canopus 310 Saturn 0.00014 a. Calculate the mean and median for these distances. b. Would the typical distance of these celestial bodies best be communicated using the mean or the median? Why? 6. Suppose that a sample of 100 homes in the metropolitan Phoenix area had a median sales price of $300,000. The mean value of these homes was $1,000,000. Explain how this could happen. Why might the median price be more informative than the mean price in describing a typical house price? 7. Suppose the mean annual income for a sample of one hundred Minneapolis residents was $50,000. Do you think the median income for this sample would have been greater than, equal to, or less than $50,000? Explain. In summary, the three measures that describe the middle of a data set are the ___________, _______________, and ___________. If a data set contains unusually large or small values, the most appropriate measure to describe a “typical” value is the ________________. Homework: (to be completed in your math workbook) 1. Using the “hours of sleep” data set from the Algebra 2/Trig survey results: a. Draw a histogram of the data set. b. Find the mean and median of the data set. c. Explain whether the median or mean best describes a typical value in the dataset. 2. Find an interesting data set online containing at least 30 values. a. Graph your data set using an appropriate graph. b. Find the mean and median of the data set. c. Explain whether the median or mean best describes a typical value in the dataset.