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Stat 281: Ch. 2--Presenting Data.

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### Stat 281: Ch. 2--Presenting Data

An engineer, consultant and statistician were driving down a steep mountain road. Suddenly, the brakes failed and the car careened down the road out of control. But half way down, the driver somehow managed to stop the car by running it against the embankment, narrowly avoiding going over a very steep cliff. They all got out, shaken, but otherwise unharmed.

The consultant said: "To fix this problem we need to organize a committee, have meetings, write several interim reports and develop a solution through a continuous improvement process."

The engineer said: "No! That would take too long, and besides that method has never really worked. I have my trusty penknife here and will take apart the brake system, isolate the problem and correct it."

The statistician said: "No - you're both wrong! Let's all push the car back up the hill and see if it happens again. We only have a sample size of 1 here!!"

The Goal

- Display data in ways that elucidate the information contained in them
- Raw Data actually contains all the information available, but it may not be easy to understand
- It’s not so much the information available that counts—it’s the information you get out!

Viewing Data Directly

- Ranked Data (aka an Array)
- Still contains all the information
- Can quickly see range (max and min)
- May also easily determine median, quartiles, etc.
- Stem and Leaf
- Arranges ranked data into chart-like form

More Complex Stem & Leaf(MiniTab Style)

Stem-and-Leaf of C1 N=16

Leaf Unit=0.010

1 59 7

4 60 148

(5) 61 02669

7 62 0247

3 63 58

1 64 3

Dot Plot for Fizzy Cola Sales

- Dot plots display vertically stacked dots for each data value.
- They tend to bring out any “clustering” behavior in the data.
- Stem & Leaf and Dot Plots begin to give us a picture of the Distribution of Data.

Summarized Data

- Frequency Tables
- Grouped or ungrouped
- Frequency Distribution
- Relative Frequency Distribution
- Bar Graphs

Histogram (Numeric Data Only)

- Pie Charts

Often used for Categorical Data

Constructing a Histogram

1. Identify the high (H) and low (L) scores. Find the range. Range = H - L.

2. Select a number of classes and a class width so that the product is a bit larger than the range.

3. Pick a starting point a little smaller than L. Count from L by the width to obtain the class boundaries. Observations that fall on class boundaries are placed into the class interval to the right.

Note:

1. The class width is the difference between the upper- and lower-class boundaries.

2. There is no best choice for class widths, number of classes, or starting points.

Terms Used With Histograms

Symmetrical: The sides of the distribution are mirror images. There is a line of symmetry.

Uniform (rectangular): Every value appears with equal frequency.

Skewed: One tail is stretched out longer than the other. The direction of skewness is on the side of the longer tail (Positively vs. negatively skewed).

J-shaped: There is no tail on the side of the class with the highest frequency.

Bimodal: The two largest classes are separated by one or more classes. Often implies two populations are sampled.

Normal: The distribution is symmetric about the mean and bell-shaped.

Left-Skewed Distribution

Ages of Nuns

Distribution of Categorical Data

Cars Sold in One Week

Day Number Sold

Monday 15

Tuesday 23

Wednesday 35

Thursday 11

Friday 12

Saturday 42

Basic Pie Chart

Cars Sold in One Week

Pie Charts focus our attention on fractions

of the whole, especially for the largest classes.

Three-D Pie Chart

Cars Sold in One Week

Three-D Pie Charts are “pretty” but can

also be used to distort the image.

Manipulating 3-D Pie Charts

Cars Sold in One Week

Changing the angle or turning the pie

may affect our perception of size.

Bar Charts for Categorical Data

Cars Sold in One Week

(Bar charts for categorical data are drawn with bars

separated, while bars in histograms touch.)

Manipulating Bar Charts

Cars Sold in One Week

Cutting off the vertical axis distorts our

perception of the differences between bars.

Manipulating Bar Charts

Cars Sold in One Week

Removal of labels on the vertical axis allows bars

to be stretched upward to hide the differences.

Hmmm…

- It is proven that the celebration of birthdays is healthy. Statistics show that people who celebrate the most birthdays become the oldest.
- In earlier times, they had no statistics, so they had to fall back on lies. (Stephen Leacock)

Measures of Central Tendency

- Statistics used to locate the middle of a set of numeric data, or where the data is clustered.
- The term average may be associated with all measures of central tendency.
- The mode for discrete data is the value that occurs with greatest frequency.
- The modal class of a histogram is the class with the greatest frequency.
- A bimodal distribution has two high-frequency classes separated by classes with lower frequencies.

The Mean

Mean: The “regular” average. The sum of all the values divided by the total number of values.

The population mean, m,(lowercase Greek mu) is the mean of all x values for the population. It is a parameter of the distribution.

We usually cannot measure m but would like to estimate its value.

The Sample Mean

The sample mean, ,(read x-bar) is the mean of all x values for the sample. It is a statistic.

The mean can be greatly influenced by outliers.

E.g. Bill Gates moves to town.

Median

Median: The value of the data that occupies the middle position when the data are ranked according to size.

The sample median (statistic) may be denoted by “x tilde”:

.

The population median (parameter), M, (uppercase Greek mu), is the data value in the middle of the population.

To find the median:

1. Rank the data.

2. Determine the depth of the median.

3. Determine the value of the median.

Mode

Mode: The mode is the value of x that occurs most frequently.

Note: If two or more values in a sample are tied for the highest frequency (number of occurrences), there is no mode.

Note: Mode, as defined here, is most applicable to categorical or discrete data. The mode for continuous data is defined differently.

Other Measures of Center

Midrange: The number midway between the maximum and minimum data values. It is found by averaging the max and min.

Midquartile: Oops, we haven’t defined quartiles yet. But this is the average of the first and third quartile instead of the max and min.

Dispersion

- How spread apart are the data?
- Two populations with the same mean can have very different distributions—would like to take measure spread somehow.
- Range (max-min)
- Values in middle are ignored
- Dispersion of middle could be very different
- Use the idea of deviation from the mean:
- MAD
- Variance
- Standard Deviation

Sums of Squares

- The sum of squared deviations is denoted by SS(x) and often called the “Sum of Squares for x.”
- There are also other notations used, including SSx and Sxx

Variance

- The Variance is the statistician’s favorite measure of dispersion, but in reports or “everyday use” the standard deviation is more commonly given.
- The Standard Deviation is the square root of the variance.
- The Variance may be thought of as the average squared deviation from the mean.
- For a sample, divide by n-1.
- For a population, divide by N.

Example: Find the variance and standard deviation for the data {5, 7, 1, 3, 8}.

Interpretation of s

- Need to get a sense of the meaning of different values of dispersion measures.
- Are units same as data or squared?
- Empirical Rule: 68%, 95%, 99.7%
- Test of Normality
- Range as estimator of s

z-Scores

- Also “standardized scores” or just “standard scores.”
- Expresses a quantity in terms of its distance from the mean in standard deviation units.

More z-Scores

The z-score measures the number of standard deviations away from the mean.

z-scores typically range from -3.00 to +3.00.

z-scores may be used to make comparisons of raw scores.

You can calculate back from z-score to raw data value by using the inverse:

Percentiles

- Values of the variable that divide a set of ranked data into 100 equal subsets.
- Each set of data has 99 percentiles.
- The kth percentile, Pk, is a value such that at most k% of the data are smaller than Pk and at most (100-k)% are larger.

Procedure for finding Pk

1. Rank the n observations, lowest to highest.

2. Compute A = (nk)/100.

3. If A is an integer:

d(Pk) = A.5 (depth)

Pk is halfway between the value of the datum in the Ath position and the value of the next datum.

If A is a fraction:

d(Pk) = B, the next largest integer.

Pk is the value of the data in the Bth position.

Some programs like Excel also do interpolation

Quartiles

- Like percentiles except dividing the data set into 4 equal subsets.
- The first quartile, Q1, is the same as the 25th percentile, and
- The third quartile, Q3, is the same as the 75th percentile.
- The second quartile is the 50th percentile, which is the median.
- Sometimes finding Q1 and Q3 is described as finding the medians of the bottom half and top half of the data, respectively.

Five Number Summary

- The Min, Q1, Median, Q3, and Max
- Indicate how the data is spread out in each quarter.
- Interquartile Range is the distance between Q1 and Q3.
- The Midquartile is the average of Q1 and Q3, another measure of central tendency.

Hmmm…

- What did the Box Plot say to the outlier?
- “Don’t you dare get close to my whisker!”

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