Standard Deviation and z Scores

Transcription

1 6.1 I am learning to use technology to calculate the variance and standard deviation of a data set calculate and understand the significance of a z score relate the positive or negative scores to their locations in a histogram develop significant conclusions about a data set Success Criteria I will know I'm successful when I can calculate the variance or standard deviation of a set of data describe what the standard deviation or variance represents about a set of data explain what a z score represents Slide down to reveal calculate a z score for a particular data value and describe its significance use standard deviation, variance, and z score to draw conclusions about a data set What are some other success criteria? The ozone layer protects Earth's surface from much of the Sun's destructive radiation. Unfortunately, the ozone layer is being destroyed, in part by chlorofluorocarbons such as coolants in old refrigerators. The ozone layer's thickness can vary significantly over periods as short as a week. What other parts of our environment are changing due to pollutants? Answer 1

2 Invesgate Standard Deviaon The table shows the thickness of the ozone layer on each day of a given week. Definition 1. Calculate the mean thickness,. 2. a) Calculate the deviation from the mean for each day. Enter the results in the third column. b) Enter the sum at the bottom of the column. c) Explain the resulting sum. 3. a) Calculate the squares of the deviations from the mean. Enter the results in the fourth column. b) Enter the sum at the bottom of the column. 4. Divide the sum of the squares by 7. This is called the variance. 5. Take the square root of the variance. This is called the standard deviation. connued on next page... Click here for the solution. Investigate Standard Deviation (continued) 6. Reflect The standard deviation is the average difference of all the measurements from the mean. What other formula does this resemble? 7. Extend Your Understanding For the previous week, the mean thickness was DU, with a standard deviation of 4.8 DU. Compare these two weeks' measurements. Click here for the solution. 2

3 Variance and Standard Deviation Formulas The variance and standard deviation of a data set allow you to determine how close the values in a distribution are to the middle of the distribution. You can calculate the variance and standard deviation of a data set using the following formulas. Click to Reveal Samples rarely contain extreme values, when compared to entire populations. As a result, the variance and standard deviation are less than would be expected. To use the sample variance and standard deviation to model a population, divide by n 1 instead of n. This slightly increases their values. Example 1 Visualizing the Spread of Heights The heights of the players on two different soccer teams are graphed in the histograms below. Both teams have a mean height of 170 cm. Team A Team B a) Which team's heights would have a greater standard deviation? Why? b) The variance of the heights on team A is What is the standard deviation? c) What would the histogram for Team A look like if the standard deviation were 6? d) What would the histogram look like if the standard deviation were 0? Click here for solution. 3

4 Example 2 Calculating Variance and Standard Deviation Slugs R Us manufactures metal slugs. Quality control technicians measure the mass of a small sample of the slugs from each run. Below are the measurements (in grams) from one such run. Use technology to answer the questions. a) Plot a histogram of the data. b) Calculate the mean and standard deviation. c) Which slugs are more than 1 standard deviation from the mean? d) What would happen to the standard deviation if every slug below a mass of 59.9 g was rejected? Click here for the Nspire solution. Click here for TI 83/84 solution. Click here for Fathom solution. Population and Sample z Score A z score indicates how many standard deviations a data value lies from the mean. Click to Reveal The formulas given earlier for standard deviation are not the most efficient way to calculate the standard deviation. More efficient formulas are given below. Click to Reveal 4

5 Example 3 Analysing z Scores The mass of each baby born in a week at Grace hospital are recorded in grams below: a) These births are a sample of the population, or a sample of the births at this hospital. Determine the sample mean and sample standard deviation. b) What is the z score of the baby with a mass of 4403 g? c) Any baby which has a mass of less than 3 kg is put on special observation. What z score corresponds to a mass of 3 kg? Click here for TI 83/84 solution. Click here for the Nspire solution. Click here for Paper/Pencil solution. Click here for the Spreadsheet solution. Invesgate Standard Deviaon The table shows the thickness of the ozone layer on each day of a given week Calculate the mean thickness, = = = = = = = The mean thickness is. 2. a) Calculate the deviation from the mean for each day. Enter the results in the third column. b) Enter the sum at the bottom of the column. c) Explain the resulting sum. The measures are evenly spread about the mean, so the sum of their deviaons is 0. Rounding errors account for the approximate value. 3. a) Calculate the squares of the deviations from the mean. Enter the results in the fourth column. b) Enter the sum at the bottom of the column. 4. Divide the sum of the squares by 7. This is called the variance. 5. Take the square root of the variance. This is called the standard deviation. connued on next page... 5

6 Invesgate Standard Deviaon (connued) 6. Reflect The standard deviation is the average difference of all the measurements from the mean. What other formula does this resemble? The standard deviaon formula is a lile bit like the formula for distance between two points, which also involves the square root of the sum of the squares of differences. 7. Extend Your Understanding For the previous week, the mean thickness was DU, with a standard deviation of 4.8 DU. Compare these two weeks' measurements. In the previous week, the mean thickness of the ozone layer was 3.5 DU thicker, and the readings were more spread out, with a standard deviaon 1.61 DU higher. Example 1 Visualizing the Spread of Heights a) Which team's heights would have a greater standard deviation? Why? Team B would have a greater standard deviaon since the data are more spread out. b) The variance of the heights on team A is What is the standard deviation? The standard deviaon is the square root of the variance. c) What would the histogram for Team A look like if the standard deviation were 6? Since 6 > 3.581, the histogram would be more spread out (like Team B). d) What would the histogram look like if the standard deviation were 0? If the standard deviaon were 0, none of the data values would deviate from the mean, so the histogram would consist of a single bar at the mean. This would happen if every player on the team was exactly the same height. 6

7 Example 2 Calculang Variance and Standard Deviaon (TI 83/84 Soluon) a) Plot a histogram of the data. Press STAT, then select 1:Edit... to enter the data. Press 2ND, STAT PLOT, then select 1: Plot 1 to set up the histogram plot. Use the arrows keys and press ENTER to select the histogram for L1 (or whatever the name of the list is) Press ZOOM, then select 9:ZoomStat to graph the histogram. b) Calculate the mean and standard deviation. Press STAT. Use the arrow keys to choose CALC, then 1:1 Var Stats. Press ENTER. The mean is and the sample standard deviaon is s = c) Which slugs are more than 1 standard deviation from the mean? Any slugs which are less than or more than are more than 1 standard deviaon from the mean. These values are indicated below: d) What would happen to the standard deviation if every slug below a mass of 59.9 was rejected? The data values and are both below the 59.9 g threshold. If these values were rejected, the standard deviaon of the remaining values would be smaller (since we removed two outliers). The new standard deviaon would be Example 2 Calculang Variance and Standard Deviaon (TI Nspire Soluon) The data are already entered in the file slug mass sample data.tns a) Plot a histogram of the data. To add a page for the histogram, press ctrl i, then select 5:Add Data & Stascs. To choose the variable to plot, press menu, then select 2:Plot Properes, 5:Add X Variable, then press enter. To change the dot plot to a histogram, press menu, then select 1:Plot Type, 3:Histogram. To adjust the interval size, press menu, then select 2:Plot Properes, 2:Histogram Properes, 2: Bin Sengs, 1:Equal Bin Width. Select the width and alignment of the first bin. b) Calculate the mean and standard deviation. Press ctrl, le arrow to change to the spreadsheet page. Select a blank formula cell, then press menu, 4:Stascs, 1:Stat Calculaons, 1:One Variable Stascs..., then press enter. Select 'slug_mass in the X1 List: field, then press enter. The mean is about g and the sample standard deviaon is about g. c) Which slugs are more than 1 standard deviation from the mean? Any slugs which are less than or more than are more than 1 standard deviaon from the mean. These values are indicated below: d) What would happen to the standard deviation if every slug below a mass of 59.9 was rejected? The data values and are both below the 59.9 g threshold. If these values were rejected, the standard deviaon of the remaining values would be smaller (since we removed two outliers). The new standard deviaon would be

8 Example 2 Calculang Variance and Standard Deviaon (Fathom Soluon) The data are already entered in the file slug mass sample data.m a) Plot a histogram of the data. Drag a New Graph onto the desktop. Drag the aribute slug_mass from the case table onto the x axis of the graph. Select Histogram from the dropdown menu in the top right corner of the graph. To adjust the intervals, double click in a blank area of the graph. Adjust the binalignment and binwidth in the Graph Inspector. b) Calculate the mean and standard deviation. Double click on the collecon box to open the inspector. Select the Measures tab. The measures have already been calculated: Hint The mean is and the sample standard deviaon is s = c) Which slugs are more than 1 standard deviation from the mean? Any slugs which are less than or more than are more than 1 standard deviaon from the mean. These values are indicated below: d) What would happen to the standard deviation if every slug below a mass of 59.9 was rejected? The data values and are both below the 59.9 g threshold. If these values were rejected, the standard deviaon of the remaining values would be smaller (since we removed two outliers). The new standard deviaon would be Example 3 Analysing z Scores (Paper and Pencil Soluon) The mass of each baby born in a week at Grace hospital are recorded in grams below: a) These births are a sample of the population (or a sample of the births at this hospital). Determine the sample mean and sample standard deviation. sample mean: sample standard deviaon: First add the squares of the measurements Then use the efficient standard deviaon formula: The mean mass is 3478 g and the sample standard deviaon is about 413 g b)what is the z score of the baby with a mass of 4403 g? c) Any baby which has a mass less than 3 kg is put on special observation. What z score corresponds to a mass of 3 kg? 8

9 Example 3 Analysing z Scores (TI 83 TM /TI 84 TM Soluon) The mass of each baby born in a week at Grace hospital are recorded in grams below: a) These births are a sample of the population (or a sample of the births at this hospital). Determine the sample mean and sample standard deviation. Press STAT, then select 1:Edit... to enter the data. Press STAT. Use the arrow keys to choose CALC, then 1:1 Var Stats. Press ENTER. The mean mass is 3478 g and the sample standard deviaon is about 413 g b) What is the z score of the baby with a mass of 4403 g. c) Any baby which has a mass less than 3 kg is put on special observation. What z score corresponds to a mass of 3 kg? Example 3 Analysing z Scores (TI Nspire TM Soluon) The data are already entered in the file Newborn Masses.tns a) These births are a sample of the population (or a sample of the births at this hospital). Determine the sample mean and sample standard deviation. Select a blank formula cell, then press menu, 4:Stascs, 1:Stat Calculaons, 1:One Variable Stascs..., press enter. then Select 'newborn_mass in the X1 List: field, then press enter. The mean mass is 3478 g and the sample standard deviaon is about 413 g b) What is the z score of the baby with a mass of 4403 g. c) Any baby which has a mass less than 3 kg is put on special observation. What z score corresponds to a mass of 3 kg? 9

10 Example 3 Analysing z Scores (Spreadsheet Soluon) The data are already entered in the file Newborn Masses.csv a) These births are a sample of the population (or a sample of the births at this hospital). Determine the sample mean and sample standard deviation. Click on an empty cell and enter the formula =AVERAGE(A2:I3) ClIck on another empty cell and enter the formula =STDEV.S(A2:I3). The mean mass is 3478 g, and the sample standard deviaon is about 413 g. b) What is the z score of the baby with a mass of 4403 g. Click on an empty cell and enter the formula =(4403 AVERAGE(A2:I3))/STDEV.S(A2:I3) The z score is c) Any baby which has a mass less than 3 kg is put on special observation. What z score corresponds to a mass of 3 kg? Click on an empty cell and enter the formula =(3000 AVERAGE(A2:I3))/STDEV.S(A2:I3) The z score is

11 Attachments slug mass sample data.ftm slug mass sample data answers.ftm marks.xls slug mass sample data.tns Newborn Masses.csv Newborn Masses answers.csv Newborn Masses.tns