Chapter 7: Summarizing and Displaying Measurement (quantitative) Data
Top Ten 4-year graduation rates for public institutions
Public Institution
Navy
Four-Year Grad Rate
86
Virgina
William and Mary
85
82
Air Force
US Military Academy
Coast Guard
79
76
75
North Carolina
Michigan
Mary Washington
James Madison
73
70
70
68
Numbers to describe data:
mode which is the most frequently observed data value (70)
median and quartiles
mean
The position of the median is found by (n+1)/2 which for our data set is (10+1)/2 = 5.5 which represents that the median is located halfway between the 5 th and 6 th observation within the ordered data. To calculate the median for our data we would take the midpoint of the 5 th and 6 th observation or for this example the midpoint between 75 and 76 which is again 75.5. The interpretation is that about 50% of the top ten 4-year grad rates were at or below75.5% .
Similarly:
The position of the quartiles can be found by the calculating the median of the median! That is, the position of the first quartile called Q1 is found by (n+1)/4 = (10+1)/4 = 2.75. For us this makes the first quartile located halfway the second and third ranked school which is between 70 and 70 making quartile 1, or Q1, 70. The interpretation is that 25% of the top ten 4-year grad rates were at or below 70%.
The position of the third quartile called Q3 is found by 3*(n+1)/4 = 3*(10+1)/4 = 8.25. For us this makes the third quartile, Q3, located halfway between the 8 th and 9 th observation which comes to halfway between 82 and 85. This makes the Q3 equal to 83.5. The interpretation is that 75% of the top ten 4-year grad rates were at or below 83.5%.
mean (sum of all values divided by the total number summed): sum is 764 divided by 10 we get mean of 76.4
variability measures: i. range (max – min = 86 – 68 = 18), ii. Interquartile Range or IQR = Q3 – Q1 = 83.5 – 70 = 13.5 The IQR represents how spread out the middle 50% of the observations are since this measure is the difference between the third and first quartile (i.e. 75% - 25% = 50%)
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1.0
0.5
0.0
2.5
2.0
1.5
iii.variance and standard deviation (or SD or Std. Dev.). These two are measures of “how spread out the values are from the mean” and are related: standard deviation is the square root of the variance. Not important to know how to calculate these but to know their meaning. A basic interpretation of the standard deviation is that it is roughly the average distance the observed values are from the mean. In our example the SD is 6.4 making the variance 6.4
2 or 41.
Five number summary: min (68), Q1 (70), Median (75.5), Q3 (83.5) and max (86)
NOTE: Sometimes these values are not possible outcomes (e.g. the mean number of children in a US household is 2.2) we do NOT round the number to a whole number (e.g. we would not round this to 2). The value is important as it tells us that on average his mean number of children is less than 3 but more than 2.
Graphing measurement data and shape:
stem and leaf
histogram
boxplot
symmetric (or bell shaped), skewed, and outliers
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Stem-and-Leaf Display: In Millions
6 8
7 00
6 8
7 003
7 3
7 5
7 569
8 2
7 6 OR 8 56
7 9
8
8 2
8 5
8 6
Histogram of Four-Year Grad Rate
3.0
68 72 76 80
Four-Year Grad Rate
84 88
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This histogram would be interpreted as right-skewed or positively skewed since the extreme observations are “pulling” or “stretching” the data to the right or in a positive direction. For such distributions we would expect the mean to be more than , or to the right, of the median. This is the case for this example as the mean of 76.4 is greater than the median of 75.5.
Building a “fence” around the data to determine extreme observations or outliers.
We can use the quartiles and IQR to build a fence around our data in order to determine if any observations in our data set can be considered extreme or outliers. This fence is built by:
Lower Fence: Q1 - 1.5*IQR
Upper Fence: Q3 + 1.5*IQR
For this data set with Q1 of 70, Q3 of 83.5, and IQR of 13.5 the fence is:
Lower: 70 - 1.5*13.5 = 70 – 20.25 = 49.75
Upper: 83.5 + 1.5*13.5 = 83.5 + 20.25 = 103.75
Looking at the data we see only none of the values lie outside this “fence” meaning that there are no outliers or extreme observations in our data.
Creating a Box Plot
A box plot is simply a representation of the median and the quartiles representing a “box” and
“whiskers” representing the fence. This plot is sometimes referred to as a box-and-whisker plot.
Below is a box plot our data. The box itself consists of Q1 and Q3 with the line within the box being the median (note how the big box goes from 70 to 43.5, the Q1 and Q3, with the line within this box being at 75.5, the median). The whiskers extend out both sides of the box but only extend to the observed value closest to the lower and upper fence without exceeding these values.
For example, the lower fence is at 49.75 but the closest observation to this value without going below it is 68; thus the whisker stops at 68. The other whisker goes to 86 since this is the closest observed value that does not exceed the upper fence of 103.75 If there were any outliers the boxplot would indicate these with an asterisk *
Boxplot of Four-Year Grad Rate
85
80
75
70
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The effect of these extreme observations, also called outliers , is greatest on the mean and standard deviation/variance since the latter uses the mean in its calculation. This effect is due to the mean taking into account the values of all observations meaning that Tiger
Woods earnings has a greater impact on the mean than it does the median (and thus less of an impact on the quartiles and IQR). The range, too, is greatly affected by outliers
Empirical Rule
When the data is symmetric or bell-shaped the use of SD is quite helpful. You would find in such instances that for data shaped this way that roughly 68% of the observations fall with +/- one SD from the mean; 95% of the observations fall within +/- two standard deviations from the mean; and almost all – 99.7% of all observations fall within +/- three standard deviations from the mean.
For example the math portion of the SAT test is typically bell-shaped with a mean of 500 and standard deviation of 100. Thus we would expect that 68% of all scores would be between 400 and 600; 95% would fall between 300 and 700; and almost all (99.7%) would fall between 200 and 800.
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