Download Standard Deviation and Interpreting Standard Deviation The Mean

Standard Deviation and Interpreting Standard Deviation
 The Mean
 Mean: The sum of the data items divided by the number of items.
∑𝑥
Mean = 𝑛
where ∑ 𝑥 represents the sum of all the data items and n represents the number of items
 The Median
 Median is the data item in the middle of each set of ranked, or ordered, data.
 To find the median of a group of data items,
1. Arrange the data items in order, from smallest to largest.
2. If the number of data items is odd, the median is the data item in the middle of the list.
3. If the number of data items is even, the median is the mean of the two middle data items.
Ex: Five employees in a manufacturing plant earn salaries of $19,700, $20,400, $21,500, $22,600 and $23,000
annually. The section manager has an annual salary of $95,000.
a. Find the median annual salary for the six people.
b. Find the mean annual salary for the six people.
Note: Why is there such a big difference between both measures of central tendency? The relatively high annual
salary of the section manager, $95,000, pulls the mean salary to a value considerably higher than the median
salary. When one or more data items are much greater than the other items, these extreme values can greatly
influence the mean. In cases like this, the median, rather than the mean, is used to summarize the incomes.
Ex:
Calculate the mean and median for birth weights and mother’s ages.
 The Mode
 Mode is the data value that occurs most often in a data set.
 If more than one data value has the highest frequency, then each of these data values is a mode.
 If no data items are repeated, then the data set has no mode.
 The Range
 Used to describe the spread of data items in a data set.
 Range: The difference between the highest and the lowest data values in a data set:
Range = highest data value – lowest data value
Ex: Honolulu’s hottest day is 89º and its coldest day is 61º. The range in temperature is: 89-61 = 28º
 Standard Deviation
 A second measure of dispersion, and one that is dependent on all of the data items, is called the
standard deviation.
 The standard deviation is found by determining how much each data item differs from the mean.
∑(𝑥−𝜇)2
 Population standard deviation: 𝜎 = √
𝑁
, where the x’s represent the data values, μ represents the
mean, and N represents the total amount of data.
Ex: Ms. Mosier measured the height of her trees growing at home. The heights of the 5 trees are listed, in
inches: 45,60,67,83,95. Find the standard deviation of the heights of Ms. Mosier’s trees.
On Your Own: The table displays the number of hurricanes in the Atlantic Ocean from 1992 to 1997. What are
the mean and standard deviation?
 Sampling
 A sample is part of a population.
 If you determine a sample carefully, the statistics for the sample can be used to make general
conclusions about the larger population.
 Suppose you want to know what percent of high school students in the US use Twitter everyday. It
likely would be impossible to get that answer from every student. So instead you select a sample of the
students (like at South Miami High) to estimate the percentage who use Twitter everyday.
 Sample Standard Deviation
 If only given a sample of a population, you can no longer compute the population standard deviation.
You only have a part of the population and therefore have to calculate the sample standard deviation
instead. It turns out that it is an unbiased estimator for the population.
∑(𝑥−𝑥̅ )2
 Sample standard deviation: 𝑠 = √
𝑛−1
, where the x’s represent the data values, 𝑥̅ represents the
sample mean, and n represents the number of data taken by the sample.
Ex: The blood alcohol concentrations of a sample of drivers involved in fatal crashes and then convicted with
jail sentences are given (based on data from the U.S. Department of Justice): 0.27, 0.17, 0.29. Find the sample
mean and sample standard deviation.
Ex:
a. The standard deviation for the normal distribution is 0.14 while the standard deviation for the skewed
distribution is 2.6. This is significantly greater than the symmetric distribution’s. Explain why this
makes sense.
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b. Which measures of center and spread would you report for the symmetric distribution? For the skewed
distribution? Explain your reasoning.
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Ex: Two fifth-grade classes have nearly identical mean scores on an aptitude test, but one class has a standard
deviation three times that of the other. All other factors being equal, which class is easier to teach, and why?
Ex: Shown below are the means and standard deviations of the yearly returns on two investments from 1926
through 2004.
a. Use the means to determine which investment provided the greater yearly return.
b. Use the standard deviation to determine which investment has the greater risk. Explain your answer.
 Interpreting and Understanding Standard Deviation
 As stated earlier, standard deviation measures the variation among values. Values close together will
yield a small standard deviation, whereas values spread farther apart will yield a larger standard
deviation.
 Many common statistics (such as human height, weight, or blood pressure) gathered from samples in the
natural world tend to have a normal distribution about their mean. A normal distribution has a
symmetric bell shape centered on the mean.
 We will develop a sense for values of standard deviations using the Empirical Rule. This only works
for normal distributions!!!
Ex: IQ scores of normal adults on the Weschler test have a bell-shaped distribution with a mean of 100 and a
standard deviation of 15. What percentage of adults have IQ scores between 55 and 145?
Ex: The table displays the number of U.S. hurricane strikes by decade from the years 1851 to 2000. Let’s say
we’re given that the mean is 17.6 and the standard deviation is 3.5. How many standard deviations from the
mean do all the values fall?
Ex:
On Your Own: For an English class, the average score on a research project was 82 and the standard deviation
of the normally distributed scores was 5. Sketch a normal curve showing the project scores and three standard
deviations from the mean.