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### Chapter 3

Averages and Variation

3.1 Measures of Central Tendency: Mode, Median, and Mean

3.2 Measures of Variation

3.3 Percentiles and Box-and-Whisker Plots

3.1 Measures of Central Tendency: Mode, Median, and Mean

Definition

The mode of a data set is the value that occurs most frequently.

5 3 7 2 4 4 2 4 8 3 4 3 4

‘4’ occurs most frequently in this data set, so it is the mode.

3.1 Measures of Central Tendency: Mode, Median, and Mean

The median is the central value of an ordered distribution.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Median =

2

How to find the median

1. Order the data from smallest to largest.

2. For an odd number of data values in the distribution,

Median = Middle data value

3. For an even number of data values in the distribution,

3.1 Measures of Central Tendency: Mode, Median, and Mean

We will be using the STAT|CALC|1Var StatsList function, which returns the results of many different calculations on the given List. But first, we must edit a list.

3.1 Measures of Central Tendency: Mode, Median, and Mean

1. Enter the data into a list.

[ENTER]

Hit STAT|EDIT|1:Edit…

[STAT] Button

Menu

Menu Item

3.1 Measures of Central Tendency: Mode, Median, and Mean

1. Enter the data into a list.

Hit STAT|EDIT|1:Edit…

If you are not in List 1 (L1), use the left arrow [◄] to move to List 1.

If List 1 is not empty, use the up arrow [▲] to move into List 1’s header (L1), which will highlight the header. Then press [CLEAR][ENTER].

WARNING: DO NOT PRESS [DEL][ENTER] INSTEAD OF [CLEAR][ENTER]!

3.1 Measures of Central Tendency: Mode, Median, and Mean

1. Enter the data into a list.

Hit STAT|EDIT|1:Edit…

Enter data from Example 2 part (a) into the list by typing each value and pressing [ENTER] after each value.

After completing the list, press [2nd][MODE] to QUIT out of the list editor.

3.1 Measures of Central Tendency: Mode, Median, and Mean

1. Enter the data into a list.

Hit STAT|EDIT|1:Edit…

Enter data into a list.

2. Hit STAT|CALC|1:1-Var Stats

[ENTER]

[STAT] Button

Menu Item

Use the right arrow [ ►] to select the CALC Menu

3.1 Measures of Central Tendency: Mode, Median, and Mean

1. Enter the data into a list.

Hit STAT|EDIT|1:Edit…

Enter data into a list.

2. Hit STAT|CALC|1:1-Var Stats

[ENTER]

[ENTER]

1-Var Stats

L1

Hit [2nd][1].

We now need to tell the function which list to process. Since our data is in L1, we will tell it to process L1.

3.1 Measures of Central Tendency: Mode, Median, and Mean

x=24.33333333

Σx=146

Σx²=3784

Sx=6.801960502

σx=6.209312003

↓n=6

¯

Example 2a

The TI-83/84 should now display

1-Var Stats

↑n=6

minX=18

Q1=19

Med=23

Q3=28

maxX=35

Use the down arrow [ ▼] to scroll to the bottom of the output.

The median is 23.

3.1 Measures of Central Tendency: Mode, Median, and Mean

We can now use the TI-83/84 to do part (b) as well. Instead of entering a new list, we will simply remove the ‘35’ from L1.

1. Hit STAT|EDIT|Edit…

2. Use the down arrow [ ▼] to scroll the highlight down to the ‘35’.

3. Hit [DEL]

4. QUIT out of the list editor.

5. Run STAT|CALC|1-Var Stats L1

6. Scroll down until you can see the result for the median, which is 19.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Remember this: You clear[CLEAR] a list; you delete[DEL] an element from a list. If you [DEL] a list, then the entire list will become unavailable in the list editor.

What to do if you accidentally delete a list

Lists can be easily restored if they are deleted. Just select STAT|EDIT|5:SetUpEditor . After you select it (either by pressing [5] or by scrolling down to it and pressing [ENTER]), it will appear in the main screen as a command waiting to be run. Press [ENTER] to run the SetUpEditor command, and your list will be restored.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Although the TI-83/84 orders lists internally when determining the median, it is sometimes convenient for us to have the list itself ordered.

The STAT|EDIT|SortA function sorts a list in ascending order (starting with the lowest value and progressing to the highest).

The STAT|EDIT|SortD function sorts a list in descending order (starting with the highest value and progressing to the lowest). We won’t have a need for SortD in this class.

3.1 Measures of Central Tendency: Mode, Median, and Mean

For example, let’s say we want to order the list given in Example 2.

Select the STAT|EDIT|SortA function.

[ENTER]

L1

)

SortA(

We will enter L1 because that is the list we want to sort.

We can now view L1 in the list editor by going to STAT|EDIT|Edit… .

3.1 Measures of Central Tendency: Mode, Median, and Mean

n + 1

Position of the middle value =

2

Formula for finding the position of the middle value

For an ordered data set of size n,

For an ordered data set of size 99, (99+1)/2 = 50, so the 50th data value is the middle value.

For an ordered data set of size 100, (100+1) / 2 = 50.5, so the 50th and 51st data values are the middle values.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Mean =

Number of entries

Definition

This is what most people think of when they hear the word ‘average’.

3.1 Measures of Central Tendency: Mode, Median, and Mean

x=80

Σx=560

Σx²=46762

Sx=18.08314132

σx=16.7417357

↓n=7

¯

Example 3

Let’s use the TI-83/84 to do Example 3.

1. Enter data into L2.

2. QUIT out of the list editor.

3. Run 1-Var Stats L2

The TI-83/84 should now display

The mean is 80.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Σx

¯

Sample mean = x =

n

Σx

Population mean = μ =

N

Definitions

Sample statistics:

Sample size = n

Population parameters:

Population size = N

3.1 Measures of Central Tendency: Mode, Median, and Mean

How to compute a 5% trimmed mean

1. Order the data from smallest to largest.

2. Delete the bottom 5% of the data and the top 5% of the data. Note: If the calculation of 5% of the number of data values does not produce a whole number, round to the nearest integer.

3. Compute the mean of the remaining 90% of the data.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Σxw

Weighted average =

Σw

Definition

where x is a data value and w is the weight assigned to that data value.

3.1 Measures of Central Tendency: Mode, Median, and Mean

How to find the weighted average using the TI83/84

To find the weighted average using the TI83/84, we must first enter both an x-List and a weight-List.

Then we use the function

1-Var Stats x-List,weight-List

In Example 4, we could enter the test scores into L5 and their respective weights into L6. Then we could run 1-Var Stats L5,L6.

3.1 Measures of Central Tendency: Mode, Median, and Mean

Definition

The range is the difference between the largest and smallest values of a data distribution.

Σ(x − x)2

¯

Sample variance = s2 =

n− 1

Σ(x − x)2

¯

Sample standard deviation = s =

n− 1

Definitions

Sample statistics:

3.2 Measures of Variation

x=6

Σx=36

Σx²=286

Sx=3.741657387

σx=3.415650255

↓n=6

¯

Example 6

Let’s enter our data into L1. If there is already data in L1, we will have to first clear the list before entering the data.

Now we can enter 1-Var Stats L1. The display should now show

The sample standard deviation is s = 3.74.

The sample variance is s2 = 3.742≈ 14.

Use the [x2] button

3.2 Measures of Variation

Σ(x − x)2

¯

Population variance = σ2 =

N

Σ(x − x)2

¯

Pop. standard deviation = σ =

N

Definitions

Population parameters:

3.2 Measures of Variation

x=6

Σx=36

Σx²=286

Sx=3.741657387

σx=3.415650255

↓n=6

¯

If the list represents a sample, then x = 6 and s = 3.74.

¯

Note

The TI-83/84 cannot know whether the data entered into a list represents a sample or a population. E.g., let’s use the data from Example 6. Running 1-Var Stats L1 would still yield

If the list represents a census, then μ= 6 and σ = 3.42.

Data can either be statistical or parametric, but NEVER BOTH.

If x and s represent the sample mean and sample standard deviation, respectively, then the sample coefficient of variation CV is defined to be

¯

s

CV =

· 100%

¯

x

σ

CV =

· 100%

μ

Definitions

If μ and σ represent the sample mean and sample standard deviation, respectively, then the sample coefficient of variation CV is defined to be

1 −

k2

Chebyshev’s theorem

For any set of data (either population or sample) and for any constant k greater than1, the proportion of data that must lie within k standard deviations on either side of the mean is at least

3.2 Measures of Variation

Results of Chebyshev’s theorem

For any set of data:

● at least 75% of the data fall in the interval μ– 2σ to μ + 2σ.

● at least 88.9% of the data fall in the interval μ– 3σ to μ + 3σ.

● at least 93.8% of the data fall in the interval μ– 4σ to μ + 4σ.

3.2 Measures of Variation

guaranteed at least 75% of data

guaranteed at least 88.9% of data

Results of Chebyshev’s theorem

μ–3σ

μ–2σ

μ–σ

μ

μ+σ

μ+2σ

μ+3σ

3.2 Measures of Variation

In practice, you can use x and s as point estimates for μ and σ in Chebyshev’s theorem.

¯

3.2 Measures of Variation

Your trifold insert does not contain Chebyshev’s theorem. It would be wise to write it in yourself as you are guaranteed to encounter it on a test.

3.2 Measures of Variation

The term ‘grouped data’ refers to data sets in the form of classes with corresponding frequencies. Frequency tables and histograms are examples of grouped data.

In practice, sometimes grouped data is the only data we have available to us. To estimate the mean and standard deviation of grouped data, we will treat the midpoints of each class as an x-List and the frequencies of each class as a weight-List, and then run the familiar 1-Var Stats x-List,weight-List.

3.2 Measures of Variation

15

13

10

8

Frequency

5

5

4

3

2

Midpoints

1

4

7

10

13

16

19

Grouped Data Example

Enter midpoints for the x-List.

Enter frequencies for the weight-List.

3.2 Measures of Variation

15

13

10

8

Frequency

5

5

4

3

2

Midpoints

1

4

7

10

13

16

19

x = 10.18, s≈ 4.43

¯

Grouped Data Example

Run 1-Var Stats x-List,weight-List

3.2 Measures of Variation

3.3 Percentiles and BoxandWhisker Plots

Definition

For whole numbers P (where 1 ≤ P ≤ 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it and (100 – P)% of the data fall at or above it.

40

30

Frequency

20

10

Score

39.5

49.5

59.5

69.5

79.5

89.5

99.5

Figure 3-3

60% of scores at or below

40% of scores at or above

60th Percentile

3.3 Percentiles and Box-and-Whisker Plots

1%

1%

1%

1%

1%

1%

1%

…

Lowest

1st

2nd

3rd

4th

5th

98th

99th

Highest

Percentiles

Figure 3-4

In an ideal situation, there are 99 percentiles which divide the data set evenly into 100 equal parts.

If the size of a data set is not divisible by 100, then the percentiles will not divide the data evenly.

We will not be concerned with such situations. Instead, we will focus on quartiles.

25%

25%

25%

Lowest

Highest

Definitions

The first quartile, Q1, is the 25th percentile.

The second quartile, Q2, is the 50th percentile, which is the same as the median.

The third quartile, Q3, is the 75th percentile.

Figure 3-5

Q1

Q2

Median

50th Percentile

Q3

3.3 Percentiles and Box-and-Whisker Plots

1. Order the data from smallest to largest.

2. Find the median. This is the second quartile Q2.

3. The first quartile Q1 is then the median of the lower half of the data; i.e., it is the median of the data falling below the Q2 position (and not including Q2).

4. The third quartile Q3 is then the median of the upper half of the data; i.e., it is the median of the data falling above the Q2 position (and not including Q2).

3.3 Percentiles and Box-and-Whisker Plots

The TI-83/84 can compute quartiles with its 1Var Stats x-List function.

3.3 Percentiles and Box-and-Whisker Plots

x=.6311111111

Σx=17.04

Σx²=14.0812

Sx=.3577207047

σx=.3510337468

↓n=27

¯

Example 9

Let’s enter our data into L3. If there is already data in L3, we will have to first clear the list before entering the data.

Now we can enter 1-Var Stats L3. The display should now show

1-Var Stats

↑n=27

minX=.16

Q1=.33

Med=.5

Q3=1

maxX=1.23

Scroll down

Let’s enter our data into L3. If there is already data in L3, we will have to first clear the list before entering the data.

Now we can enter 1-Var Stats L3. The display should now show

1-Var Stats

↑n=27

minX=.16

Q1=.33

Med=.5

Q3=1

maxX=1.23

Q1 = 0.33

Q2 = 0.50

Q3 = 1.00

IQR = Q3−Q1 =1.00 − 0.33 = 0.67

Five-number summary

Lowest value, Q1, median, Q3, highest value

3.3 Percentiles and Box-and-Whisker Plots

The 1-Var Stats function gives the five-number summary at the end of its output.

1-Var Stats

↑n=...

minX=...

Q1=...

Med=...

Q3=...

maxX=...

Five-number summary

3.3 Percentiles and Box-and-Whisker Plots

How to make a box-and-whisker plot

1. Draw a vertical scale to include the lowest and highest data values.

2. To the right of the scale, draw a box from Q1 to Q3.

3. Include a solid line through the box at the median level.

4. Draw solid lines, called whiskers, from Q1 to the lowest value and from Q3 to the highest value.

3.3 Percentiles and Box-and-Whisker Plots

Lowest value = 111;

Q1 = 182;

median = 221.5;

Q3 = 319;

highest value = 439.

Calories in Vanilla-Flavored Ice Cream Bars

450

439

400

350

319

300

250

221.5

200

182

150

111

100

3.3 Percentiles and Box-and-Whisker Plots

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