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1 Lesson Median and Mode

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2 Lesson Median and Mode California Standard: Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets. What it means for you: You’ll learn to find the median and mode for sets of data. Key words: central tendency data median mode values

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3 Lesson Median and Mode The median and mode are both intended to “summarize” a whole data set in a single number. But although they’re similar in some ways, they’re worked out very differently. This Lesson is about how to find them. They should show some kind of “most usual” or “middle” value of a set of data.

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4 Data Sets Can Contain All Sorts of Values Lesson Median and Mode Data sets often contain numerical values. {12, 8, 7, 8, 8.5, 8, 9, 6} Braces “{“ and “}” are used to show that values are grouped together in a set. Data sets can contain other types of information too. {blonde, brunette, red, blonde, black} If there are two items the same, they’re listed twice. For example, the data set below represents the hair color of five students. For example, the data set below represents the number of hours eight adults said they slept last night.

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5 Lesson Median and Mode Data sets can contain huge amounts of data, and it’s very likely that most people won’t be interested in reading every single value. So, often a value that represents a typical value for the set is used. These typical values are often referred to as measures of central tendency.

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6 The Median Is a Measure of Central Tendency Lesson Median and Mode So the median is the middle value when a set of values is put in order. If you have an odd number of values, the median is fairly easy to find. The Median The median of a data set is a value that divides the set into two equal groups — one group containing values bigger than the median, the other containing smaller values.

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7 Example 1 Solution follows… Lesson Median and Mode Find the median of the following data set: Solution First, arrange the values in order: {2, 2, 3, 4, 6, 6, 8, 9, 10} There are nine values, so the median is the fifth value. {2, 2, 3, 4, 6, 6, 8, 9, 10} The median of the data set is and four values greater than the median. There are four values less than the median... median {3, 2, 6, 8, 2, 10, 6, 4, 9}

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8 Lesson Median and Mode If there’s an even number of values, then finding the median is slightly trickier because there are two middle numbers. Here, you find the value exactly midway between the two middle numbers.

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9 {4.6, 8.9, 9, 10, 10, 14.7} Example 2 Solution follows… Lesson Median and Mode Find the median of this data set: {4.6, 8.9, 9, 10, 10, 14.7} Solution The values in this data set are already in order from least to greatest. There are six values in the set — the median lies midway between the third and fourth values. The third and fourth values are 9 and 10 — so the median is 9.5. Median = 9.5 There are three values below the median and three values above the median.

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10 Guided Practice Solution follows… Lesson Median and Mode Find the median of each of the following data sets. 1. {12, 8, 10, 19, 21, 7, 14} 2. {$101, $201, $150, $198, $300} 3. {5, 8, 3, 6, 12, 9, 5, 5, 4, 11} 4. {–6, –3, 7, 4, –2, –2, 5, 2} 5. {–2.1, 5.7, 8.1, –10.2, –100, } $198

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11 The Mode Is Another Measure of Central Tendency Lesson Median and Mode The Mode The mode of a data set is the value that occurs most often. To find the mode of a data set, look for the value that’s listed more than any other value.

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12 Example 3 Solution follows… Lesson Median and Mode Solution Find the mode of the data set: The number 2 occurs three times — this is more than any other value in the set. {17, 2, 6, 8, 2, 10, 4, 35, 10, 7, 2} So the mode is 2. {17, 2, 6, 8, 2, 10, 4, 35, 10, 7, 2}

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13 Some Data Sets Have No Modes — Others Have Many Lesson Median and Mode Data sets don’t always have one mode — as the next two examples show.

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14 Lesson Median and Mode Example 4 Find the mode of this set of data: {brown, blue, green, blue, yellow, brown, orange, white} Solution Blue and brown both appear twice. No other color appears more often. So the data set has two modes, blue and brown. Solution follows…

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15 Example 5 Solution follows… Lesson Median and Mode Solution Find the mode of this set of data: {3, 5, 19, 5, 3, 19} Each number occurs twice — no value occurs more often than the others. So this data set has no modes.

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16 Guided Practice Solution follows… Lesson Median and Mode Give the mode(s) of the following data sets. 6. {$12, $8, $7.50, $7.50, $10, $8, $8, $9.50} 7. {,,,1, } 8. {0, 1, –3, 5, 1, 0, –3, –3, 0} $8 –3 and 0 1 2

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17 Median and Mode Independent Practice Solution follows… Lesson Give the median and mode(s) of the data sets in Exercises 1–4. 1. {42, 56, 73, 64, 42} 2. {2, 5, 7, 3, 8, 10, 14} 3. {$16, $28, $20, $15} 4. {0.1, 0.4, 0.7, 0.4, 0.5, 0.7} 5. Write a set of data for which the mode and the median are the same. Median = 56, mode = 42 Median = 7, no mode Median = $18, no mode Median = 0.45, modes = 0.4 and 0.7 There are many answers, for example {3, 3, 3, 3, 3}.

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18 Median and Mode Independent Practice Solution follows… Lesson Exercises 6–7 are about Rick’s survey of car colors. 6. Rick listed the colors of the 25 cars in a parking lot. The mode for his list is blue. What does this tell you? 7. There are only 5 car colors on Rick’s list: white, red, black, blue, and green. The mode is blue. What can you say about the possible minimum and maximum number of blue cars? Explain your answer. The maximum number of blue cars is 21 (so that the other colors appear at least once). The minimum number of blue cars is 6 (since if there were any fewer, blue couldn’t be the mode). There were more blue cars than cars of any other color.

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19 Median and Mode Independent Practice Solution follows… Lesson In Exercises 8–11, find a number for each blank so the median is {42, 2, 5, 7, 36, __} 9. {4, 12, 18, __} 10. {2, 5, 14, 12, __} 11. {6, 6, 7, 9, 11, 17, 17, 20, 23, __} Any number greater than or equal to 12 13

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20 Decide whether each statement is true or false. Explain your answers. Median and Mode Independent Practice Solution follows… Lesson The median of a data set always equals one of the data values. False. For data sets with an even number of values, the median doesn’t have to be a value in the set — e.g. {1, 2, 2, 3, 3, 3} has a median Not all data sets have a median. True. A data set whose values cannot be put in order doesn’t have a median.

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21 Median and Mode Round Up Lesson These typical values beginning with “m” can get confusing. The median is the middle number when they’re arranged in order, and the mode is the most common value. There’s another similar “m” coming up in the next Lesson too — the mean.

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