Our next unit is entitled Statistics. Grades 4 and 5 provided students with an opportunity to do the pre-work necessary to understand the concepts of statistics. However, statistics, in general, are not formally introduced until grade 6. Therefore, the concepts covered in this unit may be new to students. We will begin by understanding what a statistical question is. Then, students will be introduced to various ways to analyze the data collected by asking a statistical question. Throughout this unit, students learn that data can be analyzed using center, spread, and shape. The first cluster of lessons in the unit looks at measures of center as ways to describe a data distribution. By the end of this week, students should be able to describe a set of data using the median, mean, and range. They should also be able to determine which measure of center is appropriate to use to describe a particular set of data. Ultimately students should be able to use measures of center and measures of variation to describe data distributions. The idea of choosing median vs. mean is relevant in the real world. Often times home prices are reported in the context of the median home price. Salaries for sports players might also be expressed in relation to the median salary. Grades, on the other hand, and sports statistics, are often reported using the mean. Students should understand the difference between the two measures of center to understand given statistics and also to become more informed consumers. KEY STANDARDS Apply and extend previous understandings of measurement and interpreting data. MCC6.SP.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. MCC6.SP.2.Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. MCC6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. MCC6.SP.4.Display numerical data in plots on a number line, including dot plots, histograms, and box plots. MCC6.SP.5.Summarize numerical data sets in relation to their context, such as by: MCC6.SP.5.a. Reporting the number of observations. MCC6.SP.5.b.Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. MCC6.SP.5.c.Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. MCC6.SP.5.d.Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. The purpose of this unit is to begin the study of statistics, beginning with examples of one variable (numerical) data sets, and evaluating the mean and median of these data sets. These are all ways of describing a data set numerically. Students should become aware of different methods of organizing data, beginning with frequency tables which will then lead into histograms. Students will look at the similarity between the shape of the histogram and the shape of the dot plot for the same data set. Students will set the frequency table intervals (histogram intervals) so that a skyscraper or pancake effect does not occur. Students will mark the mean and the median on the histograms so that they understand that both of these values describe the center of the data. Students should begin to use statistical language, such a “distribution” to describe the histogram of the data. Students should also get into the habit of describing the shape of the distribution as “single peaked, double peaked, roughly symmetric, or skewed.” Next, students will find the range and Mean Absolute Deviation (MAD) of the data sets. This will require a strong conceptual foundation as to what these terms represent and the use of mathematical operations, such as addition and division. Students will use whole numbers to evaluate the MAD. The range and the MAD help describe the spread of the data, which describes data sets in more detail than just using measures of center. Students will understand the importance of knowing this measure of spread, instead of only using the mean or median when describing a data set. Students will also learn about box plots (often called box and whisker plots). Box plots are a visual display of the 5 Number Summary: minimum, lower quartile Q1, median Q2, upper quartile Q3, and maximum. It should be emphasized that the “box” holds the middle 50% of the data, known as the Inter Quartile Range (IQR). Is it better to use the mean or the median. This may sound like an obscure technical question, but it really can matter. The short answer is "it depends" - to know which you should use, you must know how your data is distributed. The mean is the one to use with symmetrically distributed data; otherwise, use the median. MISCONCEPTIONS ***Students may believe all graphical displays are symmetrical. Exposing students to graphs of various shapes will show this to be false. ***Mode is remembered as the “most” and often students think this means the largest value, not “most frequent”. ***Students do not remember to put the numbers in order before finding median. ***Students assume that mean is always the best way to describe a set of data. ***Students needs to understand that mean is a redistribution of the data where mode and median are not. **Students may think that when data is “skewed to the left” that most of the data is on the left. In fact, the tail of the data is on the left and most of the data is on the right. Students confuse clustering and skewing. Here are some web sources that students can use to practice at home also! http://www.ixl.com/math/grade-6/calculate-mean-median-mode-and-range Calculate mean, median, mode, and range http://www.ixl.com/math/grade-6/interpret-charts-to-find-mean-median-mode-and-range Interpret charts to find mean, median, mode, and range http://www.ixl.com/math/grade-6/mean-median-mode-and-range-find-the-missing-number Mean, median, mode, and range: find the missing number http://www.ixl.com/math/grade-6/identify-representative-random-and-biased-samples Identify representative, random, and biased samples http://www.mathsisfun.com/data/index.html ***************************************************************************************** The following are our first three lessons, homework, and information that will be given out to go in your child's math binder. We will go through binders on Monday and make sure that all important information needed to close out Unit 2 is in their binder.
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April 2015
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