Practice problems for 6/30

The idea behind z-scores is that we can compare data from completely different numerical data sets by changing the scale to be the distance away from the average, with the new yardstick being the standard deviation. We can take z-scores and the first two pages of the notes to see how common a particular z-score is.

Example: let mux = 63.6 and sigmax = 2.5, which are the average and standard deviation for heights in inches of females.

1) What is the z-score for 67 inches? What percentage of the female population is greater than 67 inches tall?

2) What is the z-score for 64 inches? What percentage of the female population is less than 64 inches tall?


Going the other direction, we have the third page of the notes, which gives z-scores to three decimals that correspond to percentiles in the population. To find the raw score that corresponds to that percentile, use the formulas shown at the left.

3) To the nearest tenth of an inch, what is the height that corresponds to the 96th percentile in U.S. women's heights?

4) To the nearest tenth of an inch, what is the height that corresponds to the 24th percentile in U.S. women's heights?


Answer in the comments.

Representing the age at death demographics

The graphical representation of three different groups and their relative frequencies in multiple demographic categories makes the most sense in a bar chart with multiple colors. Red represents the whole population, green represents the Caucasian sub-population and blue represents the African American sub-population.

Click on the picture to see a larger version.

Preview of class for 6/29

Today, we will be discussing distributions of numerical variables. Some variables are evenly distributed through all the values, other skew to have more high values than low, some are the other way around with more low than high.


A lot of data sets have what is called normal distribution, where the most common values are the ones closest to the average, with values much higher or much lower than average being much rarer, and the farther from average those values are, the rarer they become. This curve, known as the bell-shaped curve or the normal curve or the Gaussian curve represents the normal distribution, where the high point shows the density of the values that are near average, and the lower levels at both ends represent the scarcity of values farther away from average.


One of the reasons to find the standard deviation of a set of numbers is to calculate the z-score of a raw value x. This tells us how many standard deviations a value is away from the average. Negative z-scores are for numbers below average and positive z-scores are for values above average. z(x) = 0 only when x = x-bar, the value is exactly at average. The first four pages in the class notes let us change z-scores into proportions, and using these numbers we can talk about the probability of finding values greater than some value x, or the probability of finding values between two values, call them x1 and x2

Practice problem for standard deviation

Here are two data sets, the number of wins for the teams in the American League as of end of play on Saturday, June 27, and the same statistic in the National League.

Set 1: 46, 42, 41, 41, 34, 41, 38, 36, 31, 31, 40, 40, 38, 31

When you input the data, the size of the data list is 14 and the average is 37 6/7 or 37.857...

===

Set 2: 38, 37, 38, 34, 21, 40, 41, 36, 35, 35, 35, 48, 39, 39, 32, 30

When you input this data set, the size of the set is 16 and the average is 36 1/8 or 36.125 exactly.

Round all answers to one place after the decimal.

a) What is the standard deviation for each set taken as a population, known as sigmax?

b) What is the standard deviation for each set taken as a sample, known as sx?

c) What is the significance of one set having a larger standard deviation than the other set, regardless of whether the measurement is done as a sample or a population?

Answers in the comments.

list to frequency table

Here is the list for heights in inches for all students who answered the question in our class survey, put in order from lowest to highest.


60, 60, 60, 61, 62, 62, 63, 63, 64, 65, 65, 66, 66, 66, 66, 66, 66, 66, 66, 68, 68, 69, 69, 69, 70, 70, 71, 71, 71, 72, 72, 74, 76, 77, 78

Only 35 subjects responded, so n = 35.

The frequency table reads as follows

_x____f(x)
60_____3
61_____1
62_____2
63_____2
64_____1
65_____2
66_____8
68_____2
69_____3
70_____2
71_____3
72_____2
74_____1
76_____1
77_____1
78_____1

You can check to see if you have missed any entries by finding the sum of the frequencies, which should be equal to n, which in this case is 35.

frequency table to dot plot and to stem and leaf plot

Here is our frequency table for heights in inches.

_x____f(x)
60_____3
61_____1
62_____2
63_____2
64_____1
65_____2
66_____8
68_____2
69_____3
70_____2
71_____3
72_____2
74_____1
76_____1
77_____1
78_____1

Here is a dot plot using this information.
* * * * * * *
* * * * * * *
* * * * * * *
* * * * * * *
* * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * *
_____________________________________
6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8

Here is a stem and leaf, with just two stems, 6 and 7.
6 | 000122334556666666688999
7 | 00111224678

We can make a stem and leaf where each of the stems is split into high and low, 60 to 64, 65 to 69, 70 to 74 and 75 to 79.
6 | 000122334
6 | 556666666688999
7 | 00111224
7 | 678

The idea is that the stem and leaf gives the shape of the data. It's okay to have the stem values go from low to high or high to low, as long as the values are consistent.

7 | 00111224678
6 | 000122334556666666688999

Or if we split the decades, we get this.

7 | 678
7 | 00111224
6 | 556666666688999
6 | 000122334

Five number summary and box and whiskers plot

The five number summary does not list all the information, but instead lists the low value, Q1, Q2, Q3 and the high value. The Q's stand for quartile, which means Q1 is the split for the low 25% of the data, Q2 is the median and Q3 is the split for the high half of the data. You can think of Q1 and Q2 as the medians of the low half and the high half, respectively. The interquartile range or IQR is the distance from the third quartile to the first, which is Q3 - Q1.

Let's look at the list.

7 | 678
7 | 00111224
6 | 556666666688999
6 | 000122334

With 35 entries, the middle entry is the 18th, either counting from top to bottom or bottom to top. That is the 6 marked in red and bold, so Q2 = 66.

There are 17 entries in the top half and 17 in the low half, so the position of the high and low quartiles is 9 away from the top and 9 away from the bottom, respectively. These are marked in bold and blue, with Q1 = 64 and Q3 = 71.

The five number summary is as follows

High = 78
Q3 = 71
Q2 = 66
Q1 = 64
Low = 60

Now we draw the box and whiskers in three steps, shown in the picture.


The first step is to draw the box between the values for the first and third quartiles, with a dotted line at the second quartile. We also can put dots to mark the high and low values.

The next step is computing the Interquartile Range, IQR = Q3 - Q1, which in this case is 71-64 = 7. We mark boundaries called the threshold which will be 1.5 times IQR above the third quartile and 1.5 time IQR below the first quartile. In our case, this would be at 71+ 1.5*7 = 81.5 and 64 - 1.5*7 = 63.5.

The third step is checking for outliers, If the high and low values are within these thresholds, the whiskers extend to the high value to the right and the low value to the left. (Box and whiskers can also be drawn vertically, so change the directions to high an dlow in that case.) If all the data is inside, the whiskers extend all the way to high and low. If there is data outside, those count as outliers and the whiskers are drawn to the highest data inside the high threshold and the lowest data inside the low threshold. One this has been done, we erase the thresholds.