Somerset Area School District / Overview · Web viewMeasures of Central Tendency and Dispersion...

Keystone

AlgebraCOVID-

19 Packet

#3

Student Name:___________________

Teacher:

___________________

Ms. Ritenour – Pd. 9

Mr. Flyte –

Mrs. Anderson –

Mr. DeBlase -

**Ms. Ritenour’s students: I will be holding an optional class on Zoom (either by internet or phone) every Tuesday and Thursday from 10:30a.m. – 11:00a.m. Attendance is optional and it is only supplementary to the packet. This is to provide additional explanations on the topics in the packet, if you need it. I will still be available during my office hours.

Keystone Algebra COVID-19 Packet #3 Name:________________________________

Measures of Central Tendency and Dispersion

Central Tendency : A number that represents the middle of a set of data. There are 3 measures of central tendency – Mean, Median, Mode.

o Mean – The sum of the data values divided by the number of values; the average. o Median : The number in the middle of the data when it is in order from the smallest to

the largest. Note: When there is an even set of data, there will be two values in the center so you must average the two in the middle to get the median.

o Mode – the value that occurs most often in a set of data values. No mode – every data value occurs the exact same number of times. One Mode – one value occurred more than the other values. Multiple Modes – Each mode occurred the same number of times in relation to

each other, but more than the rest of the data values. Dispersion : A measure of dispersion tells you how far the data is spread out. The Range is a

measure of dispersion. o Range : To find the range: largest value – smallest value.

Example 1 A scientist recorded the temperatures, in degrees Celsius, in 12 different parts of a rainforest. Her results were: 11, 14, 12, 15, 8, 16, 21, 10, 11, 17, 13, 10. Find the Mean, Median, Mode and Range

First, put your data in order from smallest to largest.8, 10, 10, 11, 11, 12, 13, 14, 15, 16, 17, 21

o Mean: 8+10+10+11+11+12+13+14+15+16+17+21

12=158

12=13.17

o Median: There are 2 numbers in the middle: 12, 13 so average them12+13

2=25

2=12.5

o Mode: There are two modes: 10, 11o Range: 21 – 8 = 13

Example 2: Fred is a student in Algebra. He has one more test before the end of the grading period. So far, he has earned a 85, 79, 72, and 81 on his tests. What score will he need to get on the next test if he wants to get an 83% this grading period?

o Grades are an example of an average, or a mean. Therefore, we can apply our knowledge of how to find the mean and adapt it to find the missing data value. Note, below is the problem, we will use x to represent that last test we do not know what grade he got yet. Since we know what mean he wants, we can plug that number in for the MeanMean = Sum of all data values over the total number

Mean=85+79+72+81+x5

83=85+79+72+81+x5

83=317+ x5

Fred would need a 98 on the next test to get an 83% average for the grading period.

Multiply by 5 to start solving


415=317+x98=x

Central Tendency Practice

6) Jaime's scores on his last 4 tests are shown below. 97, 89, 78, 88

Jaime has one more test in this grading period. If his goal is to have a 90% average for the grading period, what would he need to score on the last test to meet his goal?


7) Heather's scores in 7 basketball games are as follows:18, 15, 10, 11, 16, 14, 18Heather has one more game and she wants to average 15 points for all 8 games. How many points does she need to score in her last game?

8) To test the exhaust the fumes of a car, an inspector took 6 samples. The exhaust samples contained the following amounts of gas in parts per million (ppm): 8, 5, 6, 7, 9, and 5. If the maximum allowable mean is 6ppm, did the car pass the test? Explain.

9) Kelsey is keeping track of her test grades for the semester. She has had 7 tests so far and the grades are as follows:

96, 80, 98, 89, 75, 85, 96

a. What is the mean of her test scores?

b. What is the median of her test scores?

c. Kelsey wants to get a 90% for the semester. Assuming she only has one test left, what would that grade need to be for her to get a 90% for the semester?


10.

Quartiles and Interquartile Range

Quartiles and Interquartile Range help summarize data.

Quartiles break the data into four equal sections and the divisions are called Quartiles. When we say equal, we mean that the same number of data values are in the ranges bound by these three values. Quartile 1 (Q1) = The median of the lower half.Quartile 2 (Q2) = the median of the entire data set.Quartile 3 (Q3) = The median of the upper half of the data

To find the quartiles, start by finding the median of the set of data. This divides the set into 2 parts. Then, find the median of the lower half (Q1). Then, find the median of the upper half (Q3).

Interquartile Range (IQR) : the range between the first and third quartiles. This shows the middle 50% of the data.

o To find the IQR: Q3 – Q1

Example (from Example 1): A scientist recorded the temperatures, in degrees Celsius, in 12 different parts of a rainforest. Her results were: 11, 14, 12, 15, 8, 16, 21, 10, 11, 17, 13, 10. Find the first and third quartiles and the Interquartile Range.


Conclusions you can make about your data from the quartiles and interquartile range:o About 25% of data falls below the first quartile.o About 25% of the data falls above the third quartile.o About 50% of the data falls between the first and third quartiles (This is the IQR).

Practice – Complete the following 2 questions.

Review: Central Tendency and Quartiles


4.

5.


Predictions from Data and Data Displays

Data is presented in different forms of a graph depending on what the intention of the graph is trying to demonstrate. Often, you will be asked to make predictions about this data based from the graph.

o NOTE – Watch your rounding rules. If you are asked to make a prediction that involves a number of objects or a number of people, you must round to the nearest whole number. Otherwise, you may leave your answer as a decimal.

Percent Equation

The Percent Equation will be used several times to make predictions about existing data or future data. This equation is how we can make a mathematical prediction based off the given equation. The equation is:

part=% ∙ totalo The part is often the prediction (this is not always the case, sometimes you can be given

the total and asked to find the part)o The % is written as a decimal. This is found from the given set of data by taking the

number of what you want divided by the total; it can be given to you in the problem; or it could be from probability. Again, this is related to the part you are working with.

o The Total depends on the situation. For example, the total could be referring to the total in the original data set. The total could also be for a new set of data if you’re predicting what will happen in a set of data based off what you were given.

Example 1: At a local animal shelter this week, 36 animals were adopted last week and 10 of those adoptions were dogs. Based on this information, predict how many dogs will be adopted next week if they expect to process 50 adoptions.

In this problem – you’re being asked to predict the number of dogs to be adopted next week. They gave you the total for next week, 50. You’re solving for the part (number of dogs) and the percent can be found by finding what percentage of adoptions this week were dogs.


Circle Graphs

Circle graphs show a category compared to the total of 100% Categories are often reported as percentages or as number of items in each category.

Example: A breakdown of Ellie’s monthly budget is displayed in a circle graph. If she makes $2450 each month, how much does she budget for her Student Loans?

Practice: Answer each question about their given circle graph

1. Before a local election, students conducted a survey to see if residents would be in favor of a longer school day. The students surveyed 50 people and displayed the results in the following graph. If the students plan to survey an additional 250 residents, about how many residents should they expect to be undecided about a longer school day?

2. The following graph shows the results of the favorite ice cream toppings of 250 students. a. How many students voted for butterscotch?

b. How many students voted for cherry?

c. 80 more students will be polled, about how many should we expect to pick Chocolate?


3.

Bar Graph

Bar graphs display data by category, but the focus is to compare category to category. The x-axis keeps track of the categories and the y-axis allows you to see the frequency of that category.

Example At a local produce market this week, a survey asked customers to identify their favorite fruit. They graphed the weekly responses in the bar graph below.

a. Which fruit is the favorite? How do you know?

b. The researcher plans to ask another 100 people. How many people should they expect to choose apple?

c. If this trend continues for another 4 weeks, how many total people will select apple over orange?


Practice

1. A local internet provider sampled 58 of their customers and recorded the age of the internet user. The company plans to sample another 250 customers. About how many out of the new sample would the company expect to be Elderly?

2. Students in a statistics class surveyed different grade levels to see how much time they spent studying this week. If this trend continues for 5 more weeks, how many more total hours of studying would we expect seniors to do versus freshman.

Line Graphs

Line graphs are used to track a change in one category over time. Each category of the data get their own line in the graph. Predictions are made by analyzing the dat.

Practice : Answer the following questions based off the given data.

A local company tracked their profit data from 1998 to 2006.

a. What were the company profits in 2000?

b. What were the company profits in 2005?

c. Predict the amount of profit the company made in 2005.


Line Plot

A Line Plot is a graph that shows the frequency of the data on a number line.

Each symbol represents 1 item unless otherwise noted. Similar to bar graphs, a line plot shows the number of items in each category and you can find

the total by counting the number of symbols. The categories will be numerical so it can be graphed on a number line.

Example : The following line plot shows a sample of 20 teachers in a school district.

There are 105 teachers in the high school. About how many teachers are between the ages of 25 and 35?

Practice Question

In the same school district as the line plot made above, there are a total of 485 teachers in the district. Use the data to predict the number of teachers that are between the ages of 40 and 55.


Box and Whisker Plots

The Box and Whisker Plot is used to display data using the Quartiles. It allows the reader to examine the range of each quarter of data. There are 5 values that are required to make a box and whisker plot. You will need to list all 5 values when creating a plot.

o The lowest value (LV) = the leftmost whiskero Quartile 1 (Q1) = left edge of the box – the median of the lower halfo Median (Q2 or med.) = the line inside the box – median of the entire data seto Quartile 3 (Q3) = right edge of the box – median of the upper halfo The highest value (HV) = the rightmost whisker

25% of the data lies in each of the ranges (as marked above). Often, you will be given a box and whisker plot and asked questions about it. Stick to conclusions based on this knowledge as well as what we listed in the quartiles section.

Example Making a box and whisker plot. Make a box and whisker plot for the following situation.

A meteorologist is tracking the temperature in the area. The temperatures were:23, 25, 21, 19, 22, 25, 28, 32, 29, 26, 27


Example Interpreting a box and whisker plot

The following box and whisker plot represents the interest rates charged by credit card companies.

1. Estimate the median interest rate charged by credit card companies.

2. What is the range of interest rates charged by credit card companies?

3. What percentage of companies charge 20% interest rate or more?

4. About how many of the companies charge a 10% interest rate or less?

5. A credit card company charges an 8% interest rate. In their advertisement, they claim their interest rates are less than 75% of the credit card companies. Based off this data, is their claim accurate? Justify your answer.

Practice Answer the following questions about box and whisker plots

1. The accompanying box-and-whisker plot represents the cost, in dollars, of twelve CD’s.

a) Which cost is the upper quartile?

b) What is the range of the costs of the CD’s?


c) What is the median?

d) What percentage of CD’s cost more than the median?

e) How many CD’s cost between $14.50 and $26.00?

f) How many CD’s cost less than $14.50?

2. The accompanying box-and-whisker plot represents the scores earned on a math test.

2a) What is the median score?

(a) 75 (b) 70 (c) 85 (d) 77

2b) What score represents the first quartile?

(a) 55 (b) 70 (c) 100 (d) 75

2c) What statement is not true about the box and whisker plot shown?

(a) 75 represents the mean score (c) 85 represents the 3rd quartile

(b) 100 represents the maximum score (d) 55 represents the minimum score

2d) A score of an 85 on the box-and-whisker plot shown refers to:

(a) the third quartile (c) the maximum score

(b) the median (d) the mean


3. A movie theater recorded the number of tickets sold daily for a popular movie during the month of June. The box-and-whisker plot shown below represents the data for the number of tickets sold, in hundreds.

Which conclusion can be made using this plot?

(a) The second quartile is 600.

(b) The mean of the attendance is 400.

(c) The range of the attendance is 300 to 600.

(d) Twenty-five percent of the attendance is between 300 and 400.

4. The accompanying box-and-whisker plots can be used to compare the annual incomes of three professions.

Based on the box-and-whisker plots, which statement is true?

(a) The median income for nuclear engineers is greater than the income of all musicians. (b) The median income for police officers and musicians is the same.

(c) All nuclear engineers earn more than all police officers.

(d) A musician will eventually earn more than a police officer.


5. The data set 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19 represents the number of hours spent on

the Internet in a week by students in a mathematics class.

a. Create a box-and-whisker plot for this data.

b. What is the median about of hours spent on the Internet by these students?

c. The teacher of this class makes the following statement: 75% of my students spend 17 hours or less on the Internet per week. Is this teacher’s statement true or false? Explain your reasoning.

Somerset Area School District / Overview · Web viewMeasures of Central Tendency and Dispersion...

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