# 1.2: Data Summary and Presentation

- Page ID
- 5692

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Grouping Data

Suppose you were given the following data:

87, 72, 91, 91, 73, 83, 79, 81, 87, 72, 81, 91, 73, 73, 73

If you were told you were going to evaluate this data using common methods of central tendency and dispersion, how might you start by organizing the data in order to make the study as straightforward as possible?

### Grouping Data

Data in its original form, just a list of numbers, names, letters, colors, etc., is known as ** raw data**, and is often not particularly useful without some kind of organization. The series of numbers in the concept question above, for instance, doesn’t really mean anything at the moment. Without some sort of context and some level of organization, this is just a bunch of meaningless values.

Data can be classified into two general types, **quantitative** and **qualitative**. There are a number of ways to group or organize each type of data to make it more useful.

reynermedia;CK-12 Foundation -www.flickr.com/photos/892284...06/11322953266

**Quantitative**Data (data that may be conveniently described numerically):- Dates or times are commonly organized chronologically
- Data with recurring values is often organized in a
**frequency distribution** data likely to be used for evaluating the mean or the range of a population is generally organized in increasing or decreasing magnitude or alphabetically.**Univariate**data is usually organized in a table showing how the two variables change in relation to each other.**Bivariate**- Compare and contrast tables are excellent for evaluating two or more variables

**Qualitative**Data (data that may be difficult to describe with numerical values)- Commonly grouped by category
- Categories are often evaluated using a frequency distribution
- Data may be organized in order of importance
**Inductive**organization orders information by increasing complexity, listing facts prior to conclusions and advancing from specific examples to general conclusions.organization is the inverse of inductive, listing recommendations/conclusions followed by supporting facts/data.*Deductive*

### Organizing Data

Elaina is preparing to create a ** histogram** to illustrate the data that she collected on average time spent taking a particular test in her Statistics class.

16 mins, 18.5 mins, 14.5 mins, 16 mins, 19 mins, 18 mins, 16.5 mins, 15 mins, 15 mins, 14.5 mins, 14 mins, 16 mins, 12.5 mins, 19.5 mins, 14 mins, 15 mins, 16.5 mins, 14 mins, 18 mins, 16 mins

A histogram is a graph that illustrates the ** relative frequency** or

**of a single variable.**

**probability density**a. How should she organize the data to make the construction of the histogram as straightforward as possible.

Since Elaina will need to identify the number of values in each category of the data, it would be ideal to organize the data in groups called classes or intervals. With the given data, intervals of 1 minute each would seem appropriate.

b. What will the data look like after it is organized?

Minutes Required to Complete the Test:

12.5 | 14, 14, 14, 14.5, 14.5 | 15, 15, 15 | 16, 16, 16, 16, 16.5, 16.5 | 18, 18, 18.5 | 19, 19.5

Since Elaina will need to identify the number of values in each category of the data, it would be ideal to organize the data in groups called classes or intervals. With the given data, intervals of 1 minute each would seem appropriate.

### Organizing Raw Data to Create a Box-and-Whisker Plot

Orlando is planning to create a box-and-whisker plot to illustrate how much more popular dogs and cats are as pets than fish-tanks, reptiles, and birds. He has collected the data below from a randomized sample of homes in his town, using a survey questioning the number of pets each family has in each category

House 1: 2 dogs, 2 cats, 0 birds, 0 reptiles, 1 fish tank

House 2: 3 dogs, 2 cats, 1 birds, 0 reptiles, 0 fish tank

House 3: 0 dogs, 3 cats, 0 birds, 1 reptiles, 1 fish tank

House 4: 2 dogs, 1 cats, 2 birds, 0 reptiles, 1 fish tank

House 5: 2 dogs, 1 cats, 0 birds, 0 reptiles, 0 fish tank

House 6: 2 dogs, 2 cats, 0 birds, 0 reptiles, 0 fish tank

House 7: 3 dogs, 1 cats, 1 birds, 0 reptiles, 2 fish tank

House 8: 3 dogs, 2 cats, 0 birds, 0 reptiles, 1 fish tank

House 9: 2 dogs, 3 cats, 0 birds, 0 reptiles, 0 fish tank

House 10: 1 dogs, 3 cats, 0 birds, 0 reptiles, 0 fish tanks

How should Orlando organize the raw data to facilitate the creation of his box-and-whisker plot? What will the organized data look like?

Since Orlando’s box-and-whisker plot is specifically meant to highlight the number of dogs and cats, it would be a good idea to organize the data in groups by importance, with dogs and cats first. Since he will need to identify the mean, range, and quartiles of the data, it would also be good to organize each group by increasing values.

- Dogs: 0, 1, 2, 2, 2, 2, 2, 3, 3, 3
- Cats: 1, 1, 1, 2, 2, 2, 2, 3, 3, 3
- Birds: 0, 0, 0, 0, 0, 0, 0, 1, 1, 2
- Reptiles: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
- Fish Tanks: 0, 0, 0, 0, 0, 1, 1, 1, 1, 2

### Consolidating Data

Cheng is interested in the phenomenon of the changes in how fast time seems to pass to people as they age. He has collected data from 300 people between the ages of 10 and 70. Each person reported the time it * seemed* to take to complete three neutral (neither particularly liked nor disliked) activities, one 5mins, one 15mins, and one 60mins long. Now Cheng has a massive and somewhat intimidating list of numbers, and he needs to decide how to organize what he has into something useful.

With such a huge amount of raw data, Cheng’s greatest challenge will be consolidating it into a useful and informative format.

a. Identify at least 2 different ways that Cheng might organize the raw data that would illustrate changes in time perception as people age.

Cheng might choose to organize the data by increasing time in several age groups, sorting the values first by age, and then by perceived time for each activity. He might also wish to sort first by actual activity length, then by age or perceived time passage.

b. How might Cheng consolidate the data so he doesn't end up needing to plot nearly 1000 values on a chart or graph?

Finding the mean perceived length for each of several age ranges would be a great way for Cheng to maintain the general integrity of his data while reducing the sheer volume.

### Earlier Problem Revisited

*87, 72, 91, 91, 73, 83, 79, 81, 87, 72, 81, 91, 73, 73, 73*

*If you were told you were going to evaluate this data using common methods of central tendency and dispersion, what sort of preparation could you do in order to make the study as straightforward as possible?*

**Central tendency** measurements are generally facilitated by organizing data in increasing value from left to right. Ideally, it would be convenient to also note the total number of values, along with their sum, as you are ordering them.

## Examples

A class of 40 students took a science exam. They earned the following percentages on their tests:

73, 45, 62, 34, 59, 20, 48, 50, 78, 38, 52, 91, 57, 82, 46, 51, 62, 58, 39, 50, 72, 73, 63, 52, 41, 37, 28, 46, 71, 75, 36, 28, 44, 90, 51, 28, 60, 18, 47, 40.

### Example 1

Describe or demonstrate a means of displaying the results more clearly.

A good start would be to simply organize the numbers in increasing order:

18, 20, 28, 28, 34, 36, 37, 38, 39, 40, 41, 44, 45, 46, 46, 47, 48, 50, 50, 51, 51, 52, 52, 57, 58, 59, 60, 62, 62, 63, 71, 72, 73, 73, 75, 78, 82, 90, 91.

Now we can see at a glance that the numbers range from 18 to 91, with a greater frequency in the mid-range than at the extremes.

### Example 2

The teacher wants to compare the student’s scores with those of another class. Describe a means of organizing the data that would make it easy to compare the two sets of data.

To compare the scores with another class, it would be convenient to have the number of scores in each range summarized. She might either tally the number of scores between 0 and 10, then 10 and 20, and so on, or just tally the number of A’s, B’s, etc.

### Example 3

The teacher gave grades as follows:

A grade: 90 and above

B grade: 80 to 89

C grade: 70 to 79

D grade: 60 to 69

F grade: 59 and below

Make a table to show how many students achieved each grade

The table would look like this:

A | B | C | D | F |

2 | 1 | 6 | 4 | 26 |

(Either that was a frightfully difficult exam, or the students didn’t study well!)

#### Example 4

Determine if the data is qualitative or quantitative.

- The majority of the people in Asia most often wear the color red.
- A survey was done among elementary age children to discover their favorite fruit.

These are both qualitative. Neither A, nor B could be expressed as numerical data.

#### Example 5

These are the numbers of cars sold at a local dealer over the last 12 days. Create a Frequency Distribution Table. 3, 5, 1, 7, 3, 2, 8, 1, 3, 2, 6, 4.

To create a frequency distribution table for 3, 5, 1, 4, 3, 2, 2, 1, 3, 2, 5, 4, simply label the values that occur in the set across the top, and the number of occurrences of each in a 2^{nd} row beneath, either as numerals or as tally marks:

Value: | 1 | 2 | 3 | 4 | 5 |

Frequency: |
2 |
3 |
3 |
2 |
2 |

### Review

For Q’s 1-3, determine if the data is qualitative or quantitative.

1. The average temperature of a particular city is 23 degrees C.

2. Determine if the number of hours a person spends in front of a computer will affect their eye sight.

3. A random survey was done to find out the average speed of cars on a highway.

4. Which letter has the greatest frequency in the following sentence?

THE SUN ALWAYS SETS IN THE WEST.

5. Joe scored the following numbers of goals in their last twenty soccer games: 3, 0, 1, 5, 4, 3, 2, 6, 4, 2, 3, 3, 0, 7, 1, 1, 2, 3, 4, 3.

- Organize the values from smallest to greatest
- Which number had the greatest frequency?

6. The following number gives the first 31 digits of pi: 3141592653589793238462643383279

- Treating each digit as a separate unit of data, how might you organize the units to prepare for an evaluation of their frequency and range (spread)?
- How would the units appear after the organization?
- What is the frequency of the digits 3, 5, and 7?

7. A die was thrown 100 times. The frequency distribution is shown in the following table:

Roll |
Frequency |

1 | 21 |

2 | 11 |

3 | 15 |

4 | 19 |

5 | 16 |

6 |
18 |

- What is the total frequency of numbers less than 4?
- What percentage of throws of the die were higher than 5?
- How many throws scored greater than 2, but less than or equal to 5?

50 students took a test with a total of 10 possible points. The frequency distribution is shown in the following table:

Score |
Frequency |

0 | 1 |

1 | 2 |

2 | 1 |

3 | 3 |

4 | 1 |

5 | 4 |

6 | 9 |

7 | 8 |

8 | 7 |

9 | 10 |

10 | 4 |

8. If 60% is a passing score, how many students passed the test?

9. How many students scored above 80%?

10. What percentage of the students had 5 or more questions correct?

11. How many students scored greater than or equal to 4, but less than or equal to 7?

A spinner is in the shape of a regular heptagon marked with the numbers 1 to 7. Sue spun the spinner 50 times and recorded her results:

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7

12. Create a frequency table with the data.

13. Which spin had the greatest frequency?

14. Which spin had the least frequency?

A teacher decided to survey the students in her class to determine the number of siblings each of them had. The following numbers are the total number of siblings reported by each student in the class: 2, 0, 1, 0, 1, 0, 4, 3, 4, 9, 2, 1, 3, 1, 5, 1, 2, 1, 2, 4, 3, 2, 2, 6, 3, 2, 4, 2, 3, 5

15. Organize the numbers in a manner conducive to the creation of a frequency table.

16. Create a Frequency Table.

17. How many students were surveyed to collect this data?

18. How many families have 4 children or less?

## Additional Resources

PLIX: Play, Learn, Interact, Experience for

Practice for Data Summary and Presentation

Real World application with Social Skills