6.3 Standard Deviation and z-scores.notebookbartlettstp.weebly.com/uploads/2/3/0/2/23023680/unit_6_-_lesson_3... · was 158.2 DU, with a standard deviation of 4.8 DU. Compare these

6.3 Standard Deviation and zscores.notebook

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May 16, 2017

Standard Deviation and zScores6.1

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

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• 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?

Success Criteria

Slide down to reveal

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?


Answ

er

Possible answers include, but are not limited to: carbon monoxide, VOCs, sulphur oxides, nitrogen oxide, or smog in air and e‐coli, lead, or mercury in water.

Some of these quantities will vary more than others. We use measures of spread to keep track of how much these quantities vary. By doing so we can get a better handle on how much these variables are affected by pollution or human activity.


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Click here for the solution.

Invesgate Standard Deviaon

connued on next page...


The table shows the thickness of the ozone layer on each day of a given week.

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.

Definition

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 158.2 DU, with a standard deviation of 4.8 DU. Compare these two weeks' measurements.


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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.

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.

Click to Reveal

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.

a) Which team's heights would have a greater standard deviation? Why?

b) The variance of the heights on team A is 12.825. 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.


Team A Team B


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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.

d) What would happen to the standard deviation if every slug below a mass of 59.9 g was rejected?

c) Which slugs are more than 1 standard deviation from the mean?

Click here for TI83/84 solution.

Click here for the Nspire solution.

Click here for Fathom solution.

Population and Sample z‐Score

A zscore indicates how many standard deviations a data value lies from the mean.


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

Click to Reveal


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Example 3

Analysing z‐Scores

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.

c) Any baby which has a mass of less than 3 kg is put on special observation. What zscore corresponds to a mass of 3 kg?

b) What is the zscore of the baby with a mass of 4403 g?

The mass of each baby born in a week at Grace hospital are recorded in grams below:


Click here for TI83/84 solution. Click here for the Nspire solution.

Click here for Paper/Pencil solution. Click here for the Spreadsheet solution.

Solutions

Go back to the question

connued on next page...

Invesgate Standard Deviaon The table shows the thickness of the ozone layer on each day of a given week.

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.

1083

152 ‐ 154.71 = ‐2.71158 ‐ 154.71 = 3.29151 ‐ 154.71 = ‐3.71153 ‐ 154.71 = ‐1.71159 ‐ 154.71 = 4.29158 ‐ 154.71 = 3.29152 ‐ 154.71 = ‐2.71

0.03

The mean thickness is .

7.344110.824113.7641 2.924118.404110.8241 7.344171.4287

The measures are evenly spread about the mean, so the sum of their deviaons is 0. Rounding errors account for the approximate value.


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Solutions


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?

7. Extend Your Understanding For the previous week, the mean thickness was 158.2 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.

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.

Solutions


Example 1

a) Which team's heights would have a greater standard deviation? Why?

b) The variance of the heights on team A is 12.825. 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?

Visualizing the Spread of Heights

Team B would have a greater standard deviaon since the data are more spread out.

The standard deviaon is the square root of the variance.

Since 6 > 3.581, the histogram would be more spread out (like Team B).

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.


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Solutions


Example 2


d) What would happen to the standard deviation if every slug below a mass of 59.9 was rejected?


Any slugs which are less than or more than are more than 1 standard deviaon from the mean. These values are indicated below:


Calculang Variance and Standard Deviaon (TI‐83/84 Soluon)

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.

Press STAT. Use the arrow keys to choose CALC, then 1:1‐Var Stats. Press ENTER.

The mean is 60.005 and the sample standard deviaon is s = 0.069.

The data values 59.88 and 59.89 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 0.059.



Solutions


Example 2




Calculang Variance and Standard Deviaon (TI‐Nspire Soluon)

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.

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 60.005 g and the sample standard deviaon is about 0.069 g.



The data are already entered in the file slug mass sample data.tns


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Solutions


Example 2





Calculang Variance and Standard Deviaon (Fathom Soluon)

The mean is 60.005 and the sample standard deviaon is s = 0.069.




The data are already entered in the file slug mass sample data.m

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.

Double‐click on the collecon box to open the inspector. Select the Measures tab. The measures have already been calculated:

Hint In Fathom the function stdDev()

and the function sampleStdDev() both perform a sample standard deviation. For a population standard deviation, use popStdDev().

Solutions


Example 3Analysing z‐Scores (Paper and Pencil Soluon)

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.


c) Any baby which has a mass less than 3 kg is put on special observation. What zscore corresponds to a mass of 3 kg?

b)What is the zscore of the baby with a mass of 4403 g?

Use the z‐score formula.


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


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May 16, 2017

Solutions


Example 3Analysing z‐Scores (TI‐83TM/TI‐84TM Soluon)




b)What is the zscore of the baby with a mass of 4403 g.




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.

Solutions


Example 3Analysing z‐Scores (TI‐NspireTM Soluon)



b) What is the zscore of the baby with a mass of 4403 g.

The data are already entered in the file Newborn Masses.tns

Select a blank formula cell, then press menu, 4:Stascs , 1:Stat Calculaons , 1:One‐Variable Stascs..., then press enter.Select 'newborn_mass in the X1 List: field, then press enter.





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Solutions


Example 3Analysing z‐Scores (Spreadsheet Soluon)



b) What is the zscore of the baby with a mass of 4403 g.

The data are already entered in the file Newborn Masses.csv

The mean mass is 3478 g, and the sample standard deviaon is about 413 g.

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).

Click on an empty cell and enter the formula =(4403‐AVERAGE(A2:I3))/STDEV.S(A2:I3)

The z‐score is 2.24.

Click on an empty cell and enter the formula =(3000‐AVERAGE(A2:I3))/STDEV.S(A2:I3)

The z‐score is ‐1.16.

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

SMART Notebook

SMART Notebook

marks

marks

66706585617139

6061665645857

19756876657772

54536963662271

Sheet1

SMART Notebook

SMART Notebook

Mass of Babies born in a week at Grace Hospital (g),,,,,,,,3434,3069,2925,4403,3579,3292,3255,3485,31123254,3488,3880,3755,3362,3635,2885,4289,3508

SMART Notebook

Mass of Babies born in a week at Grace Hospital (g),,,,,,,,3434,3069,2925,4403,3579,3292,3255,3485,31123254,3488,3880,3755,3362,3635,2885,4289,3508,,,,,,,,,,mean,3478.333333,,,,,,,sample standard deviation,412.7500312,,,,,,,,,,,,,,,x value,z-score,,,,,,,4403,2.24025826,,,,,,,3000,-1.158893512,,,,,

SMART Notebook

SMART Notebook

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6.3 Standard Deviation and z-scores.notebookbartlettstp.weebly.com/uploads/2/3/0/2/23023680/unit_6_-_lesson_3... · was 158.2 DU, with a standard deviation of 4.8 DU. Compare these