Chapter 3 Data Description

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1 Chapter 3 Data Description

2 Section 3.1: Measures of Central Tendency Section 3.2: Measures of Variation Section 3.3: Measures of Position

3 Section 3.1: Measures of Central Tendency Definition of Average Loosely stated, the average means the center of a distribution or the most typical case. Measures of Average (a.k.a. Measures of Central Tendency) include the mean, median, mode, and midrange.

4 Definition of Statistic A statistic is a characteristic or measure obtained by using the data values from a sample. Definition of Parameter A parameter is a characteristic or measure obtained by using the data values from a specific population.

5 Definition of Mean The mean (a.k.a. arithmetic average) is the sum of the values, divided by the total number of values. The sample mean, denoted by x, is calculated by using sample data. The sample mean is a statistic. x = x1 + x2 + x3 + + xn n = Σx n where n represents the total number of values in the sample. The population mean, denoted by µ, is calculated by using all the values in the population. The population mean is a parameter. µ = x1 + x2 + x3 + + xn N = Σx N where N represents the total number of values in the population.

6 Example The data show the number of patients in a sample of six hospitals who acquired an infection while hospitalized. Find the mean

7 Definition of Median The median is the midpoint of the data array. Procedure for Finding the Median Arrange the data values in ascending order. Determine the number of values in the data set. a If n is odd, select the middle data value as the median. b If n is even, find the mean of the two middle values.

8 Example The number of police officers killed in the line of duty over the last 7 years is shown. Find the median

9 Example The number of tornadoes that have occurred in the United States over an 8-year period follows. Find the median

10 Definition of Mode The mode is the value that occurs most often in a data set. A data set that has only one value that occurs with the greatest frequency is said to be unimodal. If a data set has two values that occur with the same greatest frequency, then both values are considered to be the mode and the data set is said to be bimodal. If a data set has more than two values that occur with the same greatest frequency, then each value is used as the mode and the data set is said to be multimodal. When no data value occurs more than once, the data set is said to have no mode.

11 Example The data shows the number of licensed nuclear reactors in the United States for a recent 15-year period. Find the mode

12 Definition of Midrange The midrange is defined as the sum of the lowest and highest values in the data set divided by 2. Midrange = lowest value + highest value 2

13 Example Find the midrange of the signing bonuses (in millions of dollars) of eight NFL players for a specific year

14 Sometimes, one must find the mean of a data set in which not all values are equally represented. Definition of Weighted Mean Find the weighted mean of a variable x by multiplying each value by its corresponding weight and dividing the sum of the products by the sum if the weights. w1x1 + w2x2 + + wnxn x = = Σ (wx) w 1 + w w n Σw

15 Example A student received the grades below. Find the student s grade point average. Course Credits (w) Grade (x) English Composition I 3 A (4 pts) Introduction to Psychology 3 C (2 pts) Biology I 4 B (3 pts) Physical Education 2 D (1 pt )

16 Normal Distribution In a normal distribution, the mean, median, and mode are the same and are at the center of the distribution.

17 Right-Skewed Distribution In a right-skewed distribution, the mean is the right of the median, and the mode is the left of the median.

18 Left-Skewed Distribution In a left-skewed distribution, the mean is the left of the median, and the mode is the right of the median.

19 Section 3.2: Measures of Variation In statistics, to describe the data set accurately, statisticians must also know the measures of variation. Measures of variations include range, variance, and standard deviation.

20 Example A testing lab wishes to test two experimental brands of outdoor paint to see how long each will last before fading. The testing lab makes 6 gallons of each paint to test. Since different chemical agents are added to each group and only six cans are involved, these two groups constitute two small populations. The results (in months) are shown. Find the mean of each group. Brand A Brand B

21 Example (cont.)

22 Definition of Range The range is the highest value minus the lowest value. Range = highest value lowest value

23 Example Find the ranges for the paints in the previous example. Brand A Brand B

24 Definition of Population Variance and Standard Deviation The population variance is the average of the squares of the distance each value is from the mean. The symbol for the population variance is σ 2. The formula for the population variance is: σ 2 = Σ ( x 2) (Σx) 2 N N where x is the individual value, µ is the population mean, and N is the population size. The population standard deviation is the square root of the variance. The symbol for the population standard deviation is σ. The corresponding formula for the population standard deviation is: Σ (x 2 ) (Σx)2 N σ = N

25 Procedure Add the data values: Σx Square the sum of the data values: (Σx) 2 Square all the data values, then add them: Σ ( x 2) Plug them into the formula: σ 2 = Σ ( x 2) (Σx) 2 N N to get the population variance (N is the population size). Take the square root of the population variance to get the population standard deviation: Σ (x 2 ) (Σx)2 N σ = N

26 Example Find the variance and standard deviation for the paints. Brand A Brand B

27 Definition of Sample Variance and Standard Deviation The formula for the sample variance (denoted by s 2 ) is: s 2 = Σ ( x 2) (Σx) 2 n n 1 where x is the individual value, x is the sample mean, and n is the sample size. The formula for the sample standard deviation (denoted by s) is: Σ (x 2 ) (Σx)2 n s = n 1

28 Example The number of public school teacher strikes in Pennsylvania for a random sample of school years is shown. Find the sample variance and the sample standard deviation

29 Chebyshev s theorem applies to any distribution regardless of its shape. However, when a distribution is normal, the following statements, which make up the empirical rule, are true: Approximately 68% of the data values will fall within 1 standard deviation of the mean. Approximately 95% of the data values will fall within 2 standard deviation of the mean. Approximately 99.7% of the data values will fall within 3 standard deviation of the mean. Because the empirical rule requires that the data be approximately normal, the results are more accurate than those of Chebyshev s theorem.

30

31 Example Suppose that the scores on a national achievement exam have a mean of 480 and a standard deviation of 90. If the scores are normally distributed, then find the range for which 68%, 95%, and 99.7% of the scores will fall between.

32 Section 3.3: Measures of Position In addition to measures of central tendency and measures of variation, there are measures of position. These measures include standard scores, percentiles, and quartiles.

33 Definition of Z-Score A z-score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation. The symbol for a z-score is z. The formula is: value mean z = standard deviation For samples, the formula is: For populations, the formula is: z = x x s z = x µ σ The z-score represents the number of standard deviations that a data value falls above or below the mean.

34 Characteristics of Z-Scores If a z-score is positive, then the score is above the mean. If a z-score is zero, then the score is the same as the mean. If a z-score is negative, then the score is below the mean. When all data for a variable are transformed into z-scores, the resulting distribution will have: a mean of 0 a standard deviation of 1

35 Example A student scored a 65 on a calculus test that had a mean of 50 and a standard deviation of 10; she scored 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative positions on the two tests.

36 Definition of Percentiles Percentiles divide the data into 100 equal groups. They are symbolized by: P 1 P 2 P 3... P 99 The percentile corresponding to a given value x is computed by using the following formula: Percentile = (number of values below x) total number of values 100

37 Example The frequency distribution for the systolic blood pressure readings (in millimeters of mercury, mmhg) of 200 randomly selected college students is shown here. Construct a percentile graph. Class Boundaries Frequency Cumulative Frequency Cumulative Percent

38 Example A teacher gives a 20-point test to 10 students. The scores are shown here. Find the percentile ranks for a score of 12 and a score of

39 Procedure Arrange the data in order from lowest to highest. Substitute into the formula: c = n p 100 where n is the total number of values and p is the percentile. If c is not a whole number, round up to the next whole number. Starting at the lowest value, count over to the next number that corresponds to the rounded-up value. If c is a whole number, use the value halfway between the cth and (c + 1)st values when counting up from the lowest value.

40 Example Using the data from previous example, find the values corresponding to the 25th percentile and the 60th percentile, respectively

41 Definition of Quartiles Quartiles divide the data into four equal groups. They are symbolized by: Q 1 Q 2 Q 3 Note: Q 1 is the same as the 25th percentile Q 2 is the same as the 50th percentile (or the median) Q 3 is the same as the 75th percentile.

42 Procedure Procedure Arrange the data in order from lowest to highest. Find the median of the data values. This is the value for Q 2. Find the median of the data values that fall below Q 2. This is the value for Q 1. Find the median of the data values that fall above Q 2. This is the value for Q 3.

43 Definition of Interquartile Range (IQR) The interquartile range (IQR) is the measure of variability which uses quartiles. It is the range of the middle 50% of the data values. The IQR is the difference between the third and first quartiles. IQR = Q 3 Q 1

44 Example Find Q 1, Q 2, Q 3, and the IQR for the data set:

45 Definition of Outliers An outlier is an extremely high or an extremely low data value when compared with the rest of the data values. The mean and standard deviation of a variable can be strongly affected by outliers, so they are called nonresistant statistics. The median and IQR are less affected by outliers, so they are called resistant statistics.

46 Procedure Arrange the data in order from lowest to highest and find Q 1 and Q 3. Find the IQR: IQR= Q 3 Q 1 Multiply the IQR by 1.5. Subtract the value obtained in Step 3 from Q 1 and add the value obtained in Step 3 to Q 3. Check the data set for any data value that is smaller than Q 1 1.5(IQR) or larger than Q (IQR).

47 Example Check the following data set for outliers

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