IT 223 -- 1/12/11

Review Problems

Draw the boxplot using this information:

Additional outliers are at 33 and 105.

Ans:

                               +--++
                               |  || 
   *     *             O ------|  ||--
                               |  ||
                               +--++ 
  +---------+---------+---------+---------+
  0        50        100       150       200

What is the ideal measurement model?
Ans: actual measurement = true measurement + random error.

Use a weighted average to compute the mean of each histogram.

(a)

Bin Count

[0,1) 1

[1,2) 3

[2,3) 4

[3,4] 1

(b)

Bin Count

[0,1) 2

[1,2) 4

[2,4] 1

(c)

Bin Count

[0,1) 2

[1,2) 4

[2,2.5) 3

[2.5,3] 1

Ans: Compute the weighted average (x₁ w₁ + ... + x_n w_n) / (w₁ + ... + w_n) , where x₁ is the midpoint of the ith bin and w₁ is the number or proportion of observations in the ith bin.

Ans for 1(a): 0.5 x 1 + 1.5 x 3 + 2.5 x 4 + 3.5 x 1   18.5
              ------------------------------------- = ---- = 5.056
                          1 + 3 + 4 + 1                9

Ans for 1(b): 0.5 x 2 + 1.5 x 4 + 3.0 x 1    10
              --------------------------- =  --- = 1.429
                       2 + 4 + 1              7

Ans for 1(c).  Use percentages for bins instead of counts:

  0.5 x 20 + 1.5 x 40 + 2.25 x 30 + 2.75 x 10   165
  ------------------------------------------- = --- = 1.65.
                    20 + 40 + 30                100

What happens to x and Q2 of a dataset
- if every observation is increased by 7?
  Ans: The mean and median are both increased by 7.
- if every observation is multiplied by 3?
  Ans: The mean and median are both multiplied by 3.
- if the largest observation is increased by 1000?
  Ans: The mean will be increased by 1000 / n; the median will be unchanged (unless n = 2).
Compute the 20% and 40% trimmed means of this dataset:
How could you compute the 30% trimmed mean?
Ans: Trimming 10% of the variables off of the bottom and 10% off of the top, means omitting 1 and 94. The average of the remaining variables is 5.175.
Trimming 20% of the variables off of the bottom and the top means omitting 1 and 3 from the top and 7 and 94 from the bottom. The average of the remaining variables is 5.125.
Trimming 15% of the variables off of the botton and the top means omitting 1 and 94 and keeping 3 and 7, but giving weights of 0.5 to 3 and 7. The weighted average is The weighted average is
```
3*0.5 + 1*4 + 1*4 + 1*5 + 1*5 + 1*6 + 1*7 + 0.5*7    36
-------------------------------------------------- = -- = 5.1429
  0.5 +  1  +  1  +  1  +  1  +  1  +  1  +  0.5      7
```
An M-estimator is a weighted average where the points close to the center are given high weights and points far from the center are given low weights. Use SPSS to compute some M-estimators of the dataset in Problem 3.
Ans: Use Analyze >> Descriptive Statistics >> Explore.

Comparison of Mean and Median

The median divides a dataset in half.
If a histogram cut out of cardboard were cut at the median line, both of the resulting pieces would weigh the same.
The mean of a dataset is its center of gravity.
Find the center of gravity of a histogram cut out of cardboard.
The mean is affected more by changes in outliers than the median is affected.
The mean is pulled in the direction of the long tail of a skewed histogram, relative to the median.
If one observation is changed, the median only changes if this observation is a middle observation.

Measures of Spread

Discussion of Measures of Spread.

Analyze the NBS-10 Dataset

Use SPSS to obtain the following for the NBS-10 Dataset:
1. x and SD+
2. Histograms with three different bin widths
3. Outliers using the boxplot
4. z-scores
5. Outliers using z-scores
6. Standard error of the average

Project 2

Go over Project 2.

The Normal Distribution

Start discussing The Normal Distribution.