Statistics, Data Analysis and Probability

1SDP.1.1 Sort objects and data by common attributes and describe the categories.

1SDP.1.2 Represent and compare data (e.g., largest, smallest, most often, least often) by using pictures, bar graphs, tally charts, and picture graphs.

SDP.1.2.1 Describe, extend, and explain ways to get to a next element in simple repeating patterns (e.g., rhythmic, numeric, color, and shape).

**Comments:** The following Statistics, Data Analysis, and Probability standard is also important, but it has to be handled carefully: SDP.1.2.1 Describe, extend, and explain ways to get to a next element in simple repeating patterns (e.g., rhythmic, numeric, color, and shape). Students should never get the idea that the next term *automatically *repeats (unless they are told explicitly that it does); however, it is legitimate to ask what is the *most likely *next term. In this way students begin to learn not only the usefulness of patterns in sorting and understanding data but also careful, precise patterns of thought. Examples are sequences of colors, such as red, blue, red, blue, . . . or numbers, 1, 2, 3, 1, 2, 3, 1, 2, 3, . . . But more complex series might also be used, such as 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, . . .

2SDP.1.1 Record numerical data in systematic ways, keeping track of what has been counted.

2SDP.1.2 Represent the same data set in more than one way (e.g., bar graphs and charts with tallies).

2SDP.1.3 Identify features of data sets (range and mode).

2SDP.1.4 Ask and answer simple questions related to data representations.

2SDP.2.1 Recognize, describe, and extend patterns and determine a next term in linear patterns (e.g., 4, 8, 12 ...; the number of ears on one horse, two horses, three horses, four horses).

2SDP.2.2 Solve problems involving simple number patterns.

**Comments:** Although Standard 1.0 in the Statistics, Data Analysis, and Probability strand is important for grade two, the topics in Standard 2.0 are more important in this grade.

**2SDP.1.0 **Students collect numerical data and record, organize, display, and interpret the data on bar graphs and other representations.

**2SDP.2.0 **Students demonstrate an understanding of patterns and how patterns grow and describe them in general ways. But here, as for grade one, it is important that students distinguish between the most likely next term and *the *next term. In statistics students look for likely patterns, but in mathematics students need to know the rule that generates the pattern to determine “the” next term. As an example, given only the sequence 2, 4, 6, 8, 10, students should *not *assert that the next term is 12 but, instead, that the most likely next term is 12. For example, the series might have actually been 2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 26, 28 . . . . The ability to distinguish between what is likely and what is given promotes careful, precise thought.

Samples: 2.1 Recognize, describe, and extend patterns and determine a next term in linear patterns (e.g., 4, 8, 12 . . . ; the number of ears on one horse, two horses, three horses, four horses).

Sample problem: If there are two horses on a farm, how many horseshoes will we need to shoe all the horses? Show, in an organized way, how many horseshoes we will need for 3, 4, 5, 6, 7, 8, 9, and 10 horses.

3SDP.1.1 Identify whether common events are certain, likely, unlikely, or improbable.

3SDP.1.2 Record the possible outcomes for a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times.

3SDP.1.3 Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot).

3SDP.1.4 Use the results of probability experiments to predict future events (e.g., use a line plot to predict the temperature forecast for the next day).

Comments: The most important standards for Statistics, Data Analysis, and Probability are: **1.2 **Record the possible outcomes for a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times. **1.3 **Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot).

Samples:

**1.0 **Students conduct simple probability experiments by determining the number of possible outcomes and make simple predictions:**
**1.1 Identify whether common events are certain, likely, unlikely, or improbable.

Are any of the following certain, likely, unlikely, or impossible? 1. Take two cubes each with the numbers 1, 2, 3, 4, 5, 6 written on its six faces. Throw them at random, and the sum of the numbers on the top faces is 12. 2. It snows on New Year’s Day. 3. A baseball game is played somewhere in this country on any Sunday in July. 4. It is sunny in June. 5. Pick any two one-digit numbers, and their sum is 17.

4SDP.1.1 Formulate survey questions; systematically collect and represent data on a number line; and coordinate graphs, tables, and charts.

4SDP.1.2 Identify the mode(s) for sets of categorical data and the mode(s), median, and any apparent outliers for numerical data sets.

4SDP.1.3 Interpret one-and two-variable data graphs to answer questions about a situation.

4SDP.2.1 Represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams).

4SDP.2.2 Express outcomes of experimental probability situations verbally and numerically (e.g., 3 out of 4; 3 /4).

**Comments:** none

**Samples:**

**1.0 **Students organize, represent, and interpret numerical and categorical data and clearly communicate their findings:

The following table shows the ages of the girls and boys in a club. Use the information in the table to complete the graph for ages 9 and 10. (Adapted from TIMSS gr. 4, S-1)

Ages | Number of Girls | Number of Boys |

8 | 4 | 6 |

9 | 8 | 4 |

20 | 6 | 10 |

Nine identical chips numbered 1 through 9 are put in a jar. When a chip is drawn from the jar, what is the probability that it has an even number? (Adapted from TIMSS gr. 8, N-18)

2.2 Express outcomes of experimental probability situations verbally and numerically (e.g., 3 out of 4; 3⁄4): Royce has a bag with 8 red marbles, 4 blue marbles, 5 green marbles, and 9 yellow marbles all the same size. If he pulls out 1 marble without looking, which color is he most likely to choose? (CST released test)

5SDP.1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.

5SDP.1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.

5SDP.1.3 Use fractions and percentages to compare data sets of different sizes.

5SDP.1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.

5SDP.1.5 Know how to write ordered pairs correctly; for example, (

**Comments:** The ability to graph functions is an essential fundamental skill, and there is no doubt that linear functions are the most important for applications of mathematics. As a result, the importance of these topics can hardly be overestimated. Closely related to these standards are the following two standards from the Statistics, Data Analysis, and Probability strand: **1.4 **Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph. **1.5 **Know how to write ordered pairs correctly; for example, (*x, y*).

These standards indicate the ways in which the skills involved in the Algebra and Functions strand can be reinforced and applied.

**Samples: 1.0 **Students display, analyze, compare, and interpret different data sets, including data sets of different sizes:

1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.

Compute the mean, median, and mode of the following collection of 27 numbers: |

**Emphasis:** 1.0 1.1 1.2 1.3 **1.4 1.5**

6SDP.1.1 Compute the range, mean, median, and mode of data sets.

6SDP.1.2 Understand how additional data added to data sets may affect these computations of measures of central tendency.

6SDP.1.3 Understand how the inclusion or exclusion of outliers affects measures of central tendency.

6SDP.1.4 Know why a specific measure of central tendency (mean, median, mode) provides the most useful information in a given context.

6SDP.2.1 Compare different samples of a population with the data from the entire population and identify a situation in which it makes sense to use a sample.

6SDP.2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population.

6SDP.2.3 Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.

6SDP.2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased.

6SDP.2.5 Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims.

6SDP.3.1 Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.

6SDP.3.2 Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven).

6SDP.3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1- P is the probability of an event not occurring.

6SDP.3.4 Understand that the probability of either of two disjoint events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities.

6SDP.3.5 Understand the difference between independent and dependent events.

**Comments:** The study of statistics is more important in the sixth grade than in the earlier grades. One of the major objectives of studying this topic in the sixth grade is to give students some tools to help them understand the uses and misuses of statistics. The core standards for Statistics, Data Analysis, and Probability that focus on these goals are: **2.2 **Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population. **2.3 **Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached. **2.4 **Identify data that represent sampling errors and explain why the sample (and the display) might be biased. **2.5 **Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims. For example, if a study of computer use is focused solely on students from Fresno, the class might try to determine how valid the conclusions might be for the students in the entire state. Again, how valid would the conclusion of a study that interviewed 23 teachers from all over the state be for all the teachers in the state? These questions represent major applications of the type of precise and critical thinking that mathematics is supposed to facilitate in students.

In the sixth grade, students are also expected to become familiar with some of the more sophisticated aspects of probability. They start with the following standard: **3.1 **Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome. This strand is challenging but vitally important, not only for its use in statistics and probability but also as an illustration of the power of attacking problems systematically. The concepts in probability Standards 3.3 and 3.5 may be difficult for students to understand: **3.3 **Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if *P *is the probability of an event, 1-*P *is the probability of an event not occurring. **3.5 **Understand the difference between independent and dependent events. The topics in both standards need to be carefully introduced, and the terms must be defined. Both the concept that probabilities are measures of the likelihood that events might occur (numerical values for probabilities are usually expressed as numbers between 0 and 1) and the distinction between dependent and independent events are important for students to understand. If students can grasp the meaning of the terms, they can understand the basic points of these standards. This knowledge can help students reach accurate conclusions about statistical data.

**Samples:** **2.5 **Identify claims based on statistical data and, in simple cases, evaluate the validity of the claims.

Ex.1 Calvin has been identified as the best runner in your school because he won the fifty-yard dash at the all-schools track meet. Use the records of the track team shown in the table below to decide if Calvin is the best runner in the school. Explain your decision, using the data.

Ex.2 Soraya has been assigned to do a survey for the student council. However, she forgets to do this until the morning of the meeting, so she asks three of her best friends what kind of music they would like for a noon-time dance. Their opinions are what Soraya will report to student council. Do you think Soraya’s report is an accurate reflection of the kind of music that students want played for the noon-time dance? Explain your answer.

7SDP.1.1 Know various forms of display for data sets, including a stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data.

7SDP.1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level).

7SDP.1.3 Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set.

**Comments:** The most important of the three seventh grade standards in Statistics, Data Analysis, and Probability is this:

**1.3 **Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set. These are useful measures that students need to know well. Care should be taken to ensure that all students know the definitions, and many examples should be given to illustrate them.

**Samples:** **1.3 **Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set.

Here is a set of data for an exam in a mathematics class: | ||||||||||

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(a) Suppose there are 15 students in the class. Give a range of scores that would satisfy all the data shown above. | ||||||||||

(b) Suppose 7 students have scores ranging from 64 to 72. How many students might there be in the class? Explain. |

Emphasis: 1.0 1.1 1.2 **1.3**