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IT 403 -- Apr 15, 2016

Review Exercises

1. Approximately, what percentage of observations are in the interval [140, 450) for this density histogram?
   

   0.005 +
         |
   0.004 +           +--------+
         |           |        |
   0.003 +           |        |
         |           |        |
   0.002 +     +-----+        |
         |     |     |        |
   0.001 +     |     |        +-----------+
         |     |     |        |           |
   0.000 +-----+-----+-----+--+--+-----+--+--+
         0    100   200   300   400   500   600   

   Answer: Recall that the percent of observations in each interval is the area of the rectangle over that interval, which is computed as
   area = base * height = b * h.
   Let area [l, r] be the area of the histogram bar over the interval [l, r].
        area [140, 450] = area[140, 200] + area[200, 350] + area[350, 550].
   area[200, 350] is the entire rectangle so the percent of observations in this rectangle is
        area = base * height = (350 - 200) * 0.004 = 150 * 0.004 = 0.6 = 60%.
   However, the interval [140, 200] only uses part of the rectangle over [100, 200].
        We set up an equation to compute the proportion of the rectangle area over [140, 200]:
        area = base * height = (200 - 140) * 0.002 = 12%.
   The area of the third rectangle over the interval [350, 450] is
        area = base * height = (450 - 350) * 0.001 = 0.1 = 10%.
   The total area over [140, 450] is 0.12 + 0.6 + 0.1 = 0.82 = 82%.
   
2. What is the label for the vertical axis of the histogram in Exercise 1?
   Ans: Fraction of observations per horizontal unit. The label for the horizontal axis is horizontal units.
3. Compute the mean and median for this density histogram. To compute the mean, use weighted average. The weights are the areas of each histogram bar and the observations are the midpoints of the bars
   

   0.6  +
        |
   0.5  +                 +-------+
        |                 |       |
   0.4  +                 |       |
        |                 |       |
   0.3  +         +-------+       |
        |         |       |       |
   0.2  +         |       |       |
        |         |       |       |
   0.1  + +-------+       |       +-------+
        | |       |       |       |       |
   0.0  + +-------+-------+-------+-------+
          0       1       2       3       4
   

   Answer: the third histogram bar contains the median. (1-0)*0.1 + (2-1)*0.3 = 0.4 of the observations are to the left of the bin from 2; (1-0)*0.1 + (2-1)*0.3 + (3-2)*0.5 = 0.9 of the observations are to the left of 3. Therefore, if m is the median,
   (0.5-0.4)/(0.9-0.4) = (m-2)/(3-2)
   0.1/0.5 = (m-2)/1
   0.2 = m - 2
   m = 2.2
   
   To compute the mean, we use the weighted average:
   

   x1 w1 + ... + xn wn    0.5*0.1 + 1.5*0.3 + 2.5*0.5 + 3.5*0.1 
   -------------------- = -------------------------------------
      w1 + ... + wn               0.1 + 0.3 + 0.5 + 0.1
      
   0.05 + 0.45 + 1.25 + 0.35
   ------------------------- = 2.1
              1.0
   

4. What does a parsimonious description of a histogram mean?
   Answer: it means the histogram can be described succintly or with a very few descriptors.  In the case of a normal histogram, it can be described with only two descriptors, the sample mean and the sample standard deviation.
5. What does it mean to say that the normal distribution is ubiquitous in statistics? Ans: It means to say that the normal distribution shows up everywhere.
6. What is the ideal measurement model?
   Ans: It says that
   actual measurement = true measurement + random error
   The true measurement could be a prototype weight that does not change.  Or in the case of measuring the weights of potatos, the true measurement could be the "ideal potato" that nature is trying to produce. The random errors for the potatos are just random variations to do genetics and the environment in which the potato grows.
7. What are the official definitions of the meter, second, and kilogram?
   Ans: See the Ideal Measurement Model document.

The Standard Deviation

• The standard deviation is used to estimate the spread of the data.
• Estimators of the Spread of a Dataset: SD   SD⁺   MAD
• Practice Problems
  
  1. What happens to SD for a dataset if
     
     1. if every observation is increased by 7?
     2. if every observation is multiplied by 3?
  2. Without doing any calculations, compute the SD of this dataset:
          4   4   4   4   4
  3. Without doing any calculations, compute the SD of this dataset:
           0   0   0   0   10   10   10   10
  4. Compute the MAD of this dataset:
          20    10    15    15

z-scores

• Another name for z-scores is standardized scores.
• Consider the data vector x is defined as x₁, ... , x_n. X is the sample mean and SD+ is the sample standard deviation. The z-scores are defined as
        z_i = (x_i - X) / SD+.
• Compute z-scores using R like this:
  

  z <- (x - mean(x)) / sd(x)
  

• A z-score indicates how many standard units the observation is from the sample mean.
• z-scores can be used to define outliers:
  an extreme outlier is an observation that has a z-score greater than 3.0 or less than -3.0.
  a mild outlier is is an observation whose z-score is between 2.0 and 3.0 or between -3.0 and -2.0.
• The outliers defined by z-scores are usually different than the outliers defined by the boxplot.
• Practice Problem: compute the z-scores to classify the outliers of the NIST-10 Dataset.

Project 2

• Discuss Project 2a.

The Normal Distribution

• Use R to draw a normal density histogram:
  

  > x <- seq(-3, 3, 0.05)
  > y <- dnorm(x)
  > plot(x, y, type="l", main="Normal Density Histogram")