FOM 11-T50-Frequency Distributions, Histograms & Frequency

FOM 11 T50 – FREQUENCY DISTRIBUTION, HISTOGRAMS & FREQUENCY POLYGONS 1

© R. Ashby & R. Obayashi 2018. Duplication by permission only.

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) FREQUENCY DISTRIBUTION = an organization of data into a table (or graph) consisting of a series of

intervals of equal width.

2) INTERVAL = a range of numbers in which groups of data are placed.

3) TALLY LINE = a line representing a piece of data that is placed in the appropriate interval.

4) RANGE = a measure of the distance between the largest and smallest pieces of data.

5) HISTOGRAM = a graph of a frequency distribution comprised of a series of bars of equal width, the height of which represents the frequency of data within the corresponding interval.

6) MIDPOINT = the middle number of an interval, or, the number that represents half the distance between the upper and lower limits of an interval.

7) FREQUENCY POLYGON = a graph of a frequency distribution created by joining the midpoints of each interval in the frequency distribution table.

FREQUENCY DISTRIBUTION TABLES [5.2] I) Read the section on page 213 titled LEARN ABOUT the Math down to the bottom of the page.

A) This investigation presents data about the flow rate of the Red River in southern Manitoba and North Dekota (USA). The data was collected over 50 years. Analysis of such large amounts of data by hand can be very tedious and time consuming. In order to facilitate the analysis of such large amounts of data statisticians organize the data into a table called a FREQUENCY DISTRIBUTION TABLE. 1) A FREQUENCY DISTRIBUTION TABLE is a table composed of between 5 and 12 equally sized

intervals. A Tally Line for each piece of data is placed in the appropriate interval. Upon completion of tallying the data, the midpoint and frequency of tallies for each interval are calculated.

2) USE THESE STEPS TO CREATE A FREQUENCY DISTRIBUTION TABLE 1. List largest and smallest numbers in the data set. Round the largest number up and the smallest number

down to the same place value (both to the 10s, 100s, 1 000s or 10 000s etc.). 2. Determine the approximate range of numbers the data falls between by subtracting the smallest

rounded value from the largest rounded value. REMEMBER to include the units for the data. 3. Determine the number of intervals in which to organize the data by dividing the approximate range by

the number of intervals you want to use. REMEMBER: The number of intervals must be between 5 and 12 and the more intervals within the table the more accurate the analysis will be. NOTE: Choose a number of intervals that divides nicely into the approximate range.

4. Create a table having 5 columns titled Interval Number, Interval Range, Midpoint of Interval, Tally, and Frequency and one more row than the number of intervals.

5. Fill out the table by placing a tally line for each piece of data in the appropriate interval, etc.

3) SAMPLE PROBLEMS 1: Answer these questions. 1) Create a Frequency Distribution Table for the data in the investigation titled LEARN ABOUT the Math

on page 213 of your text. 1. List largest and smallest numbers in the data set. Round the largest number up and the smallest

number down to the same place value (i.e. both to the 1s, 10s, 100s, 1 000s or 10 000s etc.).

4578 m3/s = 4600 m3/s 159 m3/s = 100 m3/s

The numbers were rounded to the 100s in order to ensure the largest data piece will be included in the uppermost interval. This also ensures that the smallest piece of data will be included in the lowermost interval.



2. Determine the approximate range of numbers the data falls between by subtracting the smallest rounded value from the largest rounded value. REMEMBER to include the units for the data.

approximate range = 4600−100 = 4500 m3 / s

3. Determine the number of intervals in which to organize the data by dividing the approximate range by the number of intervals you want to use. REMEMBER: The number of intervals must be between 5 and 12 and the more intervals within the table the more accurate the analysis will be. NOTE: Choose a number of intervals that divides nicely into the approximate range.

4500 can be divided nicely by 5, 6, 9, 10 and 12:

4500/5 = 900 m3/s 4500/6 = 750 m3/s 4500/9 = 500 m3/s

4500/10 = 450 m3/s 4500/12 = 375 m3/s

NOTE: Five calculations have been shown so you can see how determining the number of intervals will influence the size of each interval. For this investigation, we will choose 10 intervals, each 450 m3/s in width. REMEMBER: The larger the amount of data the larger the number of intervals is required to provide accurate analysis.


INTERVAL NUMBER

FLOW RATE (m3/s)

INTERVAL MIDPOINT (m3/s)

TALLY FREQUENCY

1

2

3

4

5

6

7

8

9

10



5. Fill out the table by placing a tally line for each piece of data in the appropriate interval, etc. NOTE: BEGIN THE FIRST INTERVAL WITH THE SMALLEST ROUNDED PIECE OF DATA. THIS WILL ENSURE THAT ALL THE DATA TO BE TALLIED WILL FALL IN THE INTERVALS OF THE FREQUENCY DISTRIBUTION TABLE.

INTERVAL NUMBER

FLOW RATE (m3/s)

INTERVAL MIDPOINT (m3/s)

TALLY FREQUENCY

1 100 – 550 325 |||| |||| 10

2 550 – 1000 775 |||| || 7

3 1000 – 1450 1225 |||| ||| 8

4 1450 – 1900 1675 |||| |||| |||| 14

5 1900 – 2350 2125 |||| 5

6 2350 – 2800 2575 || 2

7 2800 – 3250 3025 ||| 3

8 3250 – 3700 3475 0

9 3700 – 4150 3925 0

10 4150 – 4600 4375 | 1

NOTE: When intervals share upper and lower limits, the upper limit is included in the interval while the lower limit is part of the previous interval. (i.e. Interval 1 has an upper limit of 550 while interval 2 has a lower limit of 550. The 550 in included in interval 1, NOT interval 2.)

2) Answer question 3 from page 222 of your text. 1. List largest and smallest numbers in the data set. Round the largest number up and the smallest


87” = 90” 63” = 60”

The numbers were rounded to the 10s in order to ensure the largest data piece will be included in the uppermost interval. This also ensures that the smallest piece of data will be included in the lowermost interval.


approximate range = 90"−60" = 30"


30 can be divided nicely by 5, 6 and 10:

30/5 = 6” 30/6 = 5” 30/10 = 3”

NOTE: Three calculations have been shown so you can see how determining the number of intervals will influence the size of each interval. For this investigation, we will choose 6 intervals, each 5” in width. This number of intervals is appropriate due to the relatively small amount of data.




INTERVAL NUMBER

TREE HEIGHT (inches)

INTERVAL MIDPOINT

(inches)

TALLY FREQUENCY

1

2

3

4

5

6

5. Fill out the table by placing a tally line for each piece of data in the appropriate interval, etc. NOTE: WE BEGIN THE FIRST INTERVAL WITH 61 INSTEAD OF THE SMALLEST ROUNDED PIECE OF DATA TO ENSURE THAT EACH INTERVAL IS THE SAME WIDTH (Use your fingers to count the size of each interval to demonstrate that each interval is 5” wide). THIS WILL ENSURE THAT ALL THE DATA TO BE TALLIED WILL FALL IN THE INTERVALS OF THE FREQUENCY DISTRIBUTION TABLE.

INTERVAL NUMBER

TREE HEIGHT (inches)

INTERVAL MIDPOINT

(inches)

TALLY FREQUENCY

1 61 – 65 63 ||| 3

2 66 – 70 68 ||| 3

3 71 – 75 73 |||| ||| 8

4 76 – 80 78 |||| |||| 9

5 81 – 85 83 |||| 5

6 86 – 90 88 || 2

NOTE: When intervals do not upper and lower limits, both the upper and lower limits are included in the interval. (i.e. The lower limit of 61 and the upper limit of 65 are included in Interval 1.)

3) Create a Frequency Distribution Table for the data in the table titled Times for Group B (s) on page 237 of your text. 1. List largest and smallest numbers in the data set. Round the largest number up and the smallest


19.7 s = 20 s 1.0 s = 0 s

The numbers were rounded to the 10s. This means the smallest number was rounded to the 0s. Rounding each number to an entire number is done to ensure the each data piece will be included in an interval.


approximate range = 20−0 = 20 s




20 can be divided nicely by 5 and 10: 20/5 = 10 s 20/10 = 2 s NOTE: Two calculations have been shown so you can see how determining the number of intervals will influence the size of each interval. For this investigation, we will choose 10 intervals, each 2 s in width. REMEMBER: The larger the amount of data the larger the number of intervals is required to provide accurate analysis.


INTERVAL NUMBER

TIMES (s) INTERVAL MIDPOINT (s)

TALLY FREQUENCY

1

2

3

4

5

6

7

8

9

10

5. Fill out the table by placing a tally line for each piece of data in the appropriate interval, etc. NOTE: BEGIN THE FIRST INTERVAL WITH THE SMALLEST ROUNDED PIECE OF DATA. THIS WILL ENSURE THAT ALL THE DATA TO BE TALLIED WILL FALL IN THE INTERVALS OF THE FREQUENCY DISTRIBUTION TABLE.

INTERVAL NUMBER

TIMES (s) INTERVAL MIDPOINT (s)

TALLY FREQUENCY

1 0 – 2 1 |||| |||| 9

2 2 – 4 3 |||| |||| 10

3 4 – 6 5 |||| 5

4 6 – 8 7 ||| 3

5 8 – 10 9 |||| 4

6 10 – 12 11 0

7 12 – 14 13 0

8 14 – 16 15 || 2

9 16 – 18 17 | 1

10 18 – 20 19 | 1



NOTE: When intervals share upper and lower limits, the upper limit is included in the interval while the lower limit is part of the previous interval. (i.e. Interval 2 has an upper limit of 4 while interval 3 has a lower limit of 4. The 4 in included in interval 2, NOT interval 3.)

3) REQUIRED PRACTICE 1: On separate sheets of grid paper, answer questions 1 and 2 given below. {Answers are on page 14}

a) Determine the approximate range of the data. b) Give two reasons to explain why the question wants you to use the number of intervals given. c) Create a frequency distribution table.

1) Use the percentages given in table titled Unit 1 Test found on page 212 to answer parts a – c above. Use 6 intervals for the frequency distribution table.

2) Use the percentages given in table titled Unit 2 Test found on page 212 to answer parts a – c above. Use 7 intervals for the frequency distribution table.

MANIPULATING THE DATA I) As powerful and useful as Statistics is, it is fairly easy to manipulate the data during its organization in order to

achieve a desired conclusion. This can be done through the number of different Frequency Distribution Tables that can be created using the same data. A given set of data can be organized into a Frequency Distribution Table using different numbers of intervals, using different starting numbers for the first interval, or by including a number shared by two intervals in the upper interval as opposed to the lower interval as is used throughout this notes package and the textbook examples. A) Study the Frequency Distribution Tables on page 215 of your text and the one created on pages 1 – 3 of this

notes package. All were created using the data found on page 213 of your text. NOTICE: that all three are composed of ten intervals. However, the Frequency Distribution Table at the top of page 215 begins at 0 m3/s and each interval is 500 m3/s. The Frequency Distribution Table at the bottom of page 215 begins at 150 m3/s and each interval is 450 m3/s. The Frequency Distribution Table created on pages 1 – 3 of this notes package begins at 100 m3/s and each interval is 450 m3/s. The differences in these three Frequency Distribution Tables result in very different frequencies, which can drastically alter the calculated statistic numbers thus impacting conclusions one can make.

HISTOGRAM [pages 216 & 218] I) A HISTOGRAM is a graph composed of a series of solid bars each of which representing the frequency of its

corresponding interval. The Vertical axis of the graph will always be the frequency of the data while the Horizontal axis always represents the interval organization of the information being analyzed. A) NOTE: When drawing a graph, be sure to label each axis appropriately and give your graph a title.

IMPORTANT HINTS AND GUIDELINES TO DRAW EFFECTIVE EASILY UNDERSTOOD GRAPHS ARE FOUND ON PAGES 11, 12 & 13 OF THIS NOTES PACKAGE.

B) NOTE: When drawing a graph, be sure to label each axis appropriately and give your graph a title.

1) USE THESE STEPS TO CREATE A HISTOGRAM 1. Make note of the largest frequency in the Frequency Distribution Table. Be sure the vertical axis of

the graph is high enough to include this number. Label this with the appropriate information. 2. Set up the horizontal axis so each interval is the same width. Label this with the appropriate

information. 3. For each interval, draw a bar to represent all the pieces of data in the interval.



2) SAMPLE PROBLEMS 2: Create a histogram for each Frequency Distribution Table created in the questions found in SAMPLE PROBLEMS 1.

1) Red River Flow Rates from 1950 – 1999

14 –

13 –

12 –

11 –

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 100 550 1000 1450 1900 2350 2800 3250 3700 4150 4600

Flow Rate (m3/s)

2) Cherry Tree Height

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 61 65 71 75 81 85

66 70 76 80 86 90 Height (inches)

Frequency Frequency



3) Times for Group B

11 –

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 2 4 6 8 10 12 14 16 18 20

Time (s)

3) REQUIRED PRACTICE 2: Answer these questions. {Answers are on page 15} Be sure you each of these questions under the appropriate Frequency Distribution Table you created in REQUIRED PRACTICE 1. 1) Use the frequency distribution table you created in question 1 of REQUIRED PRACTICE 1 to draw a

histogram. 2) Use the frequency distribution table you created in question 2 of REQUIRED PRACTICE 1 to draw a

histogram.

FREQUENCY POLYGONS [Graphs found on pages 217 & 219] I) A FREQUENCY POLYGON is a line graph created by plotting the midpoint of each interval of a Frequency

Distribution Table then connecting them with straight lines. The Vertical axis of the graph will always be the frequency of the data while the Horizontal axis always represents the midpoint of the interval organization of the information being analyzed. A) NOTE: IMPORTANT HINTS AND GUIDELINES TO DRAW EFFECTIVE EASILY UNDERSTOOD

GRAPHS ARE FOUND ON PAGES 11, 12 & 13 OF THIS NOTES PACKAGE.

B) NOTE: When drawing a graph, be sure to label each axis appropriately and give your graph a title.

1) USE THESE STEPS TO CREATE A FREQUENCY POLYGON 1. Copy the Horizontal and Vertical axes used to draw the histogram. Label this with the same

information as the histogram. 2. Plot the midpoint of each interval. 3. Connect each midpoint with a straight line. Be sure you draw a line from the intersection of the axes to

the first midpoint and a line from the last midpoint to the horizontal axis at the upper limit of the last interval.

Frequency



2) SAMPLE PROBLEMS 2: Create a Frequency Polygons for each Frequency Distribution Table created in the questions found in SAMPLE PROBLEMS 1.

1) Red River Flow Rates from 1950 – 1999

14 –

13 –

12 –

11 –

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 325 775 1225 1675 2125 2575 3025 3475 3925 4375

Flow Rate (m3/s)

2) Cherry Tree Height

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 63 68 73 78 83 88

Height (inches)

Frequency Frequency



3) Times for Group B

11 –

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 1 3 5 6 9 11 13 15 17 19

Time (s)

3) REQUIRED PRACTICE 3: Answer these questions. {Answers are on page 16} Be sure you each of these questions under the appropriate Frequency Distribution Table you created in REQUIRED PRACTICE 1. 1) Use the frequency distribution table you created in question 1 of REQUIRED PRACTICE 1 to draw a

Frequency Polygon. 2) Use the frequency distribution table you created in question 2 of REQUIRED PRACTICE 1 to draw a

Frequency Polygon.

INTERPRETING THE DATA [5.2] I) When the data has been organized into the frequency distribution table, histogram and or frequency polygon, it

can be analyzed. The analysis will reveal information that can be helpful for people to make decisions regarding the data they collected and studied. Read section 5.2 of your text (pages 213 – 220) to learn how to draw conclusions from the data you’ve organized. A) Entering the data in a frequency distribution table into the STAT program of a TI-83/4 calculator

1) USE THESE STEPS TO USE THE STAT PROGRAM OF YOUR CALCULATOR TO ANALYZE A DATA

ORGANIZED IN A FREQUENCY DISTRIBUTION TABLE

1) Press the STAT button. 2) Press the 1 button to enter a data set into a list. 3) Enter the midpoint of each interval in a list (i.e. L1). 4) Enter the frequency of each interval in a different list (i.e. L2). Be sure that the midpoint and

frequency of each interval are entered in the same order in each list!! 5) When the midpoint and frequency of each interval have been entered press the 2nd button then MODE

to QUIT this list. 6) Press the STAT button. 7) Press the ► button so the CALC is surrounded in black.

Frequency



8) Press the 1 button to select 1: 1-Var Stats, then the 2nd button, then NUMBER of the midpoint list, then , , then 2nd then NUMBER of the frequency list the list for you wish to analyze. i.e. press the 2nd button then 1 the data in L1 (the midpoint list), press the 2nd button then 2 the data in L2 (the frequency list) then press then ENTER to analyze the data.

9) List the mean, x = # , standard deviation, σ = #, (listed as σx = = #), median (listed as Med = #). 10) List the n = sample size = number of pieces of data in the data set. 11) Calculate the range by completing this calculation: range = maxX −minX NOTE: The calculator does not calculate the mode, you must determine it.

A) REQUIRED PRACTICE 4: 1) Pages 221 - 224: Questions 1, 2, 3, 4, 5,6, 7, 8, 9 & 10. {Answers are on pages 546 - 548 of the text} 2) Pages 239 - 240: Questions 1, 2, 3, 4, 5 & 6. {Answers are on page 549 of the text}



HOW TO DRAW BAR GRAPHS When drawing Histograms or Bar Graphs you must follow these five guidelines:

1) The graph and each axis must have a TITLE. 2) Use the same spacing between your numbers and categories. 3) The bars should be the same width (thickness). 4) USE A RULER AND BE NEAT. 5) USE AS MUCH OF THE PAPER AS POSSSIBLE.

HOW TO DRAW DOUBLE BAR GRAPHS When drawing Double Bar Graphs you must follow these seven guidelines:

1) The graph and each axis must have a TITLE. 2) Use the same spacing between your numbers and categories. 3) The bars should be the same width (thickness). 4) USE A RULER AND BE NEAT. 5) USE AS MUCH OF THE PAPER AS POSSSIBLE. 6) Use a legend to identify the two different bars. 7) Use colour or some other way to clearly differentiate the two different bars.

1) Graph Title 2) The axis has evenly Spaced Numbers (They all go up by 10).

2) + 3) Evenly Spaced Categories, and same size bar separated by even distances.

1) Axis Title 1) Axis Title

1) Bar Titles where appropriate.



HOW TO DRAW LINE GRAPHS When drawing Line Graphs you must follow these five or seven guidelines:

1) The graph and each axis must have a TITLE. Include the units in brackets. 2) Use the same spacing between your numbers. 3) Use big dots as your points. 4) TIME is ALMOST ALWAYS on the horizontal axis. 4) USE A RULER AND BE NEAT. 5) USE AS MUCH OF THE PAPER AS POSSSIBLE.

FOR DOUBLE LINE GRAPHS 6) Use a legend to identify the two different bars. 7) Use colour or some other way to clearly differentiate the two different bars.

6) Legend, use two different colors.

1) Graph Title

6) Legend, use two different colors

2) Evenly Spaced Numbers (They all go up by 1).

2) Evenly Spaced Numbers (They all go up by 1).

1) Axis Title (Units in brackets)

1) Axis Title

c) Different colours to differentiate the bars.



NOTE

ANSWERS TO THE REQUIRED PRACTICE Required Practice 1 from page 6 1a) approximate range = 85%−55% = 30% 1b) Six intervals was used because that number divides nicely into the approximate range of 30% and it is a reasonable number of intervals given the relatively small sample size. 1c)

2a) approximate range = 100%−30% = 70% 2b) Seven intervals was used because that number divides nicely into the approximate range of 70% and it is a reasonable number of intervals given the relatively small sample size. 2c)

INTERVAL NUMBER

PERCENT INTERVAL MIDPOINT PERCENT

TALLY FREQUENCY

1 55 – 60 57.5 || 2

2 60 – 65 62.5 ||| 3

3 65 – 70 67.5 ||| 3

4 70 – 75 72.5 |||| |||| | 11

5 75 – 80 77.5 |||| 5

6 80 – 85 82.5 | 1

INTERVAL NUMBER

PERCENT INTERVAL MIDPOINT PERCENT

TALLY FREQUENCY

1 30 – 40 35 | 1

2 40 –50 45 || 2

3 50 – 60 55 |||| 4

4 60 – 70 65 || 2

5 70 – 80 75 |||| |||| 9

6 80 – 90 85 |||| 4

7 90 – 100 95 ||| 3

You can use this zigzag symbol to jump from zero to another starting number. NOTE: However that after this, each number goes up by the same amount



Required Practice 2 from page 8

1) Percentages for Unit 1 Test

11 –

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 55 60 65 70 75 80 85

Percentage


10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 30 40 50 60 70 80 90 100

Percentage

Frequency Frequency



Required Practice 3 from page 10


11 –

10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 57.5 62.5 67.5 72.5 77.5 82.5 87.5

Percentage


10 –

9 –

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 0 35 45 55 65 75 85 95

Height (inches)

Frequency Frequency



ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER

LAST then FIRST Name T50 – FREQUENCY DISTRIBUTION, HISTOGRAMS & FREQUENCY POLYGONS Block:

Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!!! (Marks indicated in italicized brackets.)

REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!!

∞ Answer these questions. Be sure you show all work necessary to answer the questions.

1) The table given below lists the marks for the final exam of a full Foundations of Mathematics 11 class of 30 students. Use it to answer the questions below it.

% on the final exam 0 94 42 65 50 80

61 66 71 96 85 18 100 28 62 12 87 62 51 78 48 92 46 90 41 98 68 72 64 66

a) Create a Frequency Distribution Table having 10 intervals. Be sure you show all the calculations

required to create the table. (5)

b) Draw a Histogram and Frequency Polygon for the table. (7)

c) Which range of percentages was the most frequent? (1)

d) Create a Frequency Distribution Table having a different number of intervals (Your choice). Be sure you show all the calculations required to create the table. (5)

e) Draw a Histogram and Frequency Polygon for the table you created in question 1d). (7)

f) Which range of percentages was the most frequent for your answers to questions 1d & e)? (1)

g) How did changing the number of intervals alter the data and its analysis? (2)

∞ In this section you will analyze the data collected for two different sized bags of sunflower seeds. The data for each bag is organized into individual tables that are found at the bottom of page 228 of your text. Complete the questions below for each table. Be sure you show all work necessary to answer the questions.

a) Create a Frequency Distribution Table having 5 intervals. Be sure you show all the calculations required to create the table. (5 ea)

b) Draw a Histogram. (4 ea)

c) Use the STAT function of your calculator to determine the mean and the standard deviation. (2 ea)

2) Analysis for the 227 g bags of sunflowers.

3) Analysis for the 454 g bags of sunflowers.

4) Which size bag has the most accurately recorded mass? Justify your answer. (3)

5) List at least two problems with the Frequency Distribution tables. Explain why they are problems. (4) /57

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FOM 11-T50-Frequency Distributions, Histograms & Frequency