Math 20-2

Statistics: Lesson #3

Measures of Dispersion

Objective: By the end of the lesson, you should be able to:

-Explain the meaning of standard deviation.

-Calculate the standard deviation of a set of data, using technology.

-Solve problems involving standard deviation.

Warm-Up:

Tun Bolts Inc. produces screws and nut bolts. One of the most common nut bolts they produce for large industry commercial use has a circular opening of diameter 3 cm. The company is checking two of its machines to see if they are producing bolts correctly. Ten bolts are sampled from each machine, and their diameters (in cm) are shown below:

Machine 1 / Machine 2
2.99 2.98 3.00 3.01 3.00
3.01 2.99 3.02 3.00 2.98 / 3.01 3.03 2.98 2.96 3.02
3.02 3.00 3.01 2.97 2.99

Determine the average diameter of the bolts and the range for each machine. Do not round the answers.

Machine 1:Machine 2:

Key Point:

  • Dispersion refers to how spread out a set of data is.

-If every data value is the same, the dispersion is 0.

-As the data values get more spread out, the dispersion gets higher.

-Dispersion measures the consistency of the data.

One measure of dispersion is the range.

e.g. 1) Consider the bolts made by the two machines in the Warm-Up question.

a) Which set has the greater dispersion?

b) Which machine is making better bolts? Explain your answer.

Another measure of dispersion is the______, denoted by the Greek letter .

-Deviation refers to the difference between a data value and the mean. For example, if you got 71% on a test with an average of 66%, the deviation of your score would be:

-The standard deviation assigns a numerical value to how scattered a set of data is in relation to the mean.

-A standard deviation of 0 means that every data value is ______.

-A low standard deviation means that most of the data is ______the mean.

-A high standard deviation means that the data values are ______from the mean.

The formula for standard deviation is quite complex, so we are going to use the graphing calculator to find standard deviation.

Steps to find Standard Deviation:

  1. Press STAT
  2. Choose 1: Edit
  3. Enter all the data values into L1. Remember to press ENTER after every value, including the last one.
  4. Press STAT again.
  5. Use the right arrow key to move to the CALC menu on the top of the screen.
  6. Choose 1: 1-Var Stats
  7. Press ENTER on the main screen.

This gives a list of statistical measures, including:

-mean ()

-standard deviation ()

-lowest data value (minX)

-median (Med)

-highest data value (maxX)

e.g. 2)The coach of a varsity girls’ basketball team keeps statistics on all the players. Near the end of one game, the score is tied and the best starting guard has fouled out. The coach needs to make a substitution. The coach examines the field goal stats for three guards on the bench in the last 10 games.

a)Use the graphing calculator to find the mean, standard deviation, and range for each player’s stats. Round to the nearest hundredth, if necessary.

b)Which player has the highest mean field goal percent?

c)What patterns do you notice about the range and the standard deviation?

d)Which player is the most consistent shooter? Explain.

e)If you were the coach, which player would you substitute into the game? Explain your decision.

It is always more accurate to use raw data when calculating measures of central tendency and measures of dispersion. However, if data is given only in a frequency table, we can still use the graphing calculator to find an estimate of the mean and standard deviation, as follows:

  1. Go to the lists: Press STAT 1:Edit
  2. Enter the midpoint of each interval into L1.
  3. Enter the frequency into L2.
  4. Press STAT ► CALC 1:1-Var Stats.
  5. Before pressing ENTER, type in L1, L2 – this tells the calculator to use both lists as a frequency table.

e.g. 3) Janessa conducted a survey to determine the number of hours per week that grade 11 students inher school played video games. She organized her results for the females in this frequency table:

Gaming Hours per Week for Grade 11 Females
Hours / Frequency
0-3 / 11
3-6 / 7
6-9 / 15
9-12 / 10
12-15 / 5

a)How many grade 11 girls did Janessa survey?

b)What is the mode for this data?

c)Use your calculator to determine the mean and standard deviation of this data. Round to the nearest hundredth.

d) Janessa also surveyed grade 11 males on the same question. She determined that the mean number of gaming hours per week for males was 11.84 h, with a standard deviation of 2.06 h. Compare the results of the two surveys.

Assignment:p. 261-265 #3a, 4-9, 11, 13