# Describing Distributions with Numbers

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Describing Distributions with Numbers
Chapter 1 Section 1.2 Describing Distributions with Numbers

Parameter - Fixed value about a population Typical unknown

Value calculated from a sample
Statistic - Value calculated from a sample

Measures of Central Tendency
parameter Mean - the arithmetic average Use m to represent a population mean Use to represent a sample mean statistic This is on the formula sheet, so you do not have to memorize it. Formula: S is the capital Greek letter sigma – it means to sum the values that follow

Measures of Central Tendency
Median - the middle of the data; 50th percentile Observations must be in numerical order Is the middle single value if n is odd The average of the middle two values if n is even NOTE: n denotes the sample size

Measures of Central Tendency
Mode – the observation that occurs the most often Can be more than one mode If all values occur only once – there is no mode Not used as often as mean & median

Measures of Central Tendency
Range- The difference between the largest and smallest observations. This is only one number! Not 3-8 but 5

2 3 4 8 12 The median is 4 lollipops!
Suppose we are interested in the number of lollipops that are bought at a certain store. A sample of 5 customers buys the following number of lollipops. Find the median. The median is 4 lollipops! The numbers are in order & n is odd – so find the middle observation.

Suppose we have sample of 6 customers that buy the following number of lollipops. The median is …
The numbers are in order & n is even – so find the middle two observations. The median is 5 lollipops! Now, average these two values. 5

Suppose we have sample of 6 customers that buy the following number of lollipops. Find the mean.
To find the mean number of lollipops add the observations and divide by n.

2 3 4 6 8 20 5 7.17 The median is . . . The mean is . . .
What would happen to the median & mean if the 12 lollipops were 20? 5 The median is . . . 7.17 The mean is . . . What happened?

2 3 4 6 8 50 5 12.17 The median is . . . The mean is . . .
What would happen to the median & mean if the 20 lollipops were 50? 5 The median is . . . 12.17 The mean is . . . What happened?

Resistant - YES NO Statistics that are not affected by outliers
Is the median resistant? YES Is the mean resistant? NO

YES Look at the following data set. Find the mean.
Now find how each observation deviates from the mean. What is the sum of the deviations from the mean? Will this sum always equal zero? This is the deviation from the mean. YES

Look at the following data set. Find the mean & median.
27 27 Use scale of 2 on graph Create a histogram with the data. (use x-scale of 2) Then find the mean and median. Look at the placement of the mean and median in this symmetrical distribution.

Look at the following data set. Find the mean & median.
28.176 25 Use scale of 2 on graph Create a histogram with the data. (use x-scale of 8) Then find the mean and median. Look at the placement of the mean and median in this right skewed distribution.

Create a histogram with the data. Then find the mean and median.
Look at the following data set. Find the mean & median. Mean = Median = 54.588 58 Use scale of 2 on graph Create a histogram with the data. Then find the mean and median. Look at the placement of the mean and median in this skewed left distribution.

Go to java view

Recap: In a symmetrical distribution, the mean and median are equal.
In a skewed distribution, the mean is pulled in the direction of the skewness. In a symmetrical distribution, you should report the mean! In a skewed distribution, the median should be reported as the measure of center!

Quartiles Arrange the observations in increasing order and locate the median M in the ordered list of observations. The first quartile Q1 is the median of the 1st half of the observations The third quartile Q3 is the median of the2nd half of the observations.

median Q3 Q1 34 25 41

What if there is odd number?
median When dividing data in half, forget about the middle number

The interquartile range (IQR)
The distance between the first and third quartiles. IQR = Q3 – Q1 Always positive

Outlier: Q1=25 Q3=41 IQR=41-25=16 25 - 1.5 x 16 = 1 41 + 1.5 x 16 = 65
We call an observation an outlier if it falls more than 1.5 x IQR above the third or below the first. Let’s look back at the same data: Q1=25 Q3=41 IQR=41-25=16 x 16 = 1 x 16 = 65 Lower Cutoff Upper Cutoff

Since 73 is above the upper cutoff, we will call it an outlier.

Five-number summary Minimum Q1 Median Q3 Maximum

If you plot these five numbers on a graph, we have a ………
Boxplot

convenient handling of outliers construction is not subjective (like histograms) Used with medium or large size data sets (n > 10) useful for comparative displays

does not retain the individual observations should not be used with small data sets (n < 10)

How to construct find five-number summary Min Q1 Med Q3 Max
draw box from Q1 to Q3 draw median as center line in the box extend whiskers to min & max

ALWAYS use modified boxplots in this class!!!
display outliers fences mark off the outliers whiskers extend to largest (smallest) data value inside the fence ALWAYS use modified boxplots in this class!!!

Modified Boxplot Q1 – 1.5IQR Q3 + 1.5IQR
Interquartile Range (IQR) – is the range (length) of the box Q3 - Q1 Q1 – 1.5IQR Q IQR These are called the fences and should not be seen. Any observation outside this fence is an outlier! Put a dot for the outliers.

Modified Boxplot . . . Draw the “whisker” from the quartiles to the observation that is within the fence!

A report from the U.S. Department of Justice gave the following percent increase in federal prison populations in 20 northeastern & mid-western states in 1999. Create a modified boxplot. Describe the distribution. Use the calculator to create a modified boxplot.

Evidence suggests that a high indoor radon concentration might be linked to the development of childhood cancers. The data that follows is the radon concentration in two different samples of houses. The first sample consisted of houses in which a child was diagnosed with cancer. Houses in the second sample had no recorded cases of childhood cancer. (see data on note page) Create parallel boxplots. Compare the distributions.

Cancer No Cancer 100 200 Radon The median radon concentration for the no cancer group is lower than the median for the cancer group. The range of the cancer group is larger than the range for the no cancer group. Both distributions are skewed right. The cancer group has outliers at 39, 45, 57, and The no cancer group has outliers at 55 and 85.

Assignment 1.2

Variability

Why is the study of variability important?
Allows us to distinguish between usual & unusual values In some situations, want more/less variability scores on standardized tests time bombs medicine

Measures of Variability
range (max-min) interquartile range (Q3-Q1) deviations variance standard deviation Lower case Greek letter sigma

Suppose that we have these data values:
Find the mean. Find the deviations. What is the sum of the deviations from the mean?

Square the deviations: Find the average of the squared deviations:

The average of the deviations squared is called the variance.
parameter statistic Population Sample

Calculation of variance of a sample
df

A standard deviation is a measure of the average deviation from the mean.

Calculation of standard deviation

Degrees of Freedom (df)
n deviations contain (n - 1) independent pieces of information about variability

Which measure(s) of variability is/are resistant?
IQR

Activity (worksheet)

Linear transformation rule
When multiplying or adding a constant to a random variable, the mean and median changes by both. When multiplying or adding a constant to a random variable, the standard deviation changes only by multiplication. Formulas:

An appliance repair shop charges a \$30 service call to go to a home for a repair. It also charges \$25 per hour for labor. From past history, the average length of repairs is 1 hour 15 minutes (1.25 hours) with standard deviation of 20 minutes (1/3 hour). Including the charge for the service call, what is the mean and standard deviation for the charges for labor?

Rules for Combining two variables
To find the mean for the sum (or difference), add (or subtract) the two means To find the standard deviation of the sum (or differences), ALWAYS add the variances, then take the square root. Formulas: If variables are independent

Bicycles arrive at a bike shop in boxes
Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the times for each setup phase are independent with the following means & standard deviations (in minutes). What are the mean and standard deviation for the total bicycle setup times? Phase Mean SD Unpacking 3.5 0.7 Assembly 21.8 2.4 Tuning 12.3 2.7

Assignment 1.2B