Chapter 3. Data Description

Transcription

1 Chapter 3. Data Description

2 Graphical Methods Pie chart It is used to display the percentage of the total number of measurements falling into each of the categories of the variable by partition a circle. Bar chart To use the height of bars to represent the number of observations in categories. Frequency histogram To divide the range of the measurements by the approximate number of class intervals desired. To construct the frequency table which lists the number of the measurements, i.e., frequency, in the class intervals, and then use histograms or bars to represent the frequencies. Class histogram versus relative frequency histogram

3 Statistical Distribution (1) Probability The chance of an event occurring Probability Distribution The probability distribution of a discrete random variable is a table, graph, formula, or other device used to specify all possible values of a discete random variable along with their respective probability. Unimodal distribution A histogram with one major peak is called unimodal distribution. Bimodal distribution A histogram with two major peaks is called bimodal distribution.

4 Statistical Distribution (2) Uniform distribution Every interval has the same number of observations. Normal distribution The relative frequency histogram or probability distribution is a smooth bell-shaped curve. Lognormal distribution After taking logarithm of the values of a random variable, the relative frequency histogram shows a smooth-bell-shaped curve.

5 Exploratory Data Analysis The aim of the data analysis is to explore and understand the characteristics of the data. The basic tool for this analysis is graphical techniques. Stem-and-leaf plot It is a clever, simple device for constructing a histogramlike picture of a frequency distribution.

6 Describing Data on a Single Variable Measures of Central Tendency To describe the center of the distribution of measurements. Measures of variability To describe how the measurements vary about the center of the distribution. Parameter versus statistic Parameters are the numerical descriptive measures for a population. Statistics are the numerical descriptive measures for a sample.

7 Measures of Central Tendency (1) Mode The mode of a set of measurements is defined to be the measurement that occurs most often (with the highest frequency). The mode is also commonly used as a measure of popularity that reflects central tendency or opinion. Median The middle value when the measurements are arranged from lowest to highest. For an event number of measurements, the median is the average of the two middle values when the measurements are arranged from the lowest to highest. When there are an odd number number of measurements, the median is the middle value.

8 Measures of Central Tendency (2) Grouped data median For grouped data, the median is calculated as: median = L + w f m (0.5n cf b ) L = Lower class limit of the interval that contains the median n = total frequency cf f b m = the sum of frequencies (cumulative frequency) for all classes before the median class = frequency of the class interval containing the median w = interval width

9 Measures of Central Tendency (3) Arithmetic mean (or mean) The sum of the measurements divided by the total number of measurements. Population mean Sample mean y = Grouped data mean y n i= 1 n µ y i i= = n y y + y + y n f n i y i y f i i y n = midpoint of the i - th class interval = frequency of the i - th class interval n = the total number of measurements

10 Measures of Central Tendency (4) Trimmed mean a variation of the mean It drops the highest and lowest extreme values and averages the rest. 5% trimmed mean drops the highest 5% and the lowest 5% of the measurements and averages the rest. In a limiting sense the median is a 50% trimmed mean. Outlier the extreme values of measurements The mean is subject to distortion due to the presence pf one or more outliers.

11 Measures of Central Tendency (5) Relationship among the mean( µ ), the trimmed mean( TM ), the median( M d) and the mode( M o) for distributions with different skewness. It is not restricted to using only one measure of central tendency. For some data sets, it will be necessary to use more than one of these measures to provide an accurate descriptive summary of central tendency for the data.

12 Major Characteristics of Each Measure of Central Tendency (1) Mode It is the most frequency or probable measurement in the data set. There can be more than one mode for a data set. It is not influenced by extreme measurements. Modes of subsets cannot be combined to determine the mode of the complete data set. For grouped data its value can change depending on the categories used. It is applicable for both qualitative and quantitative data.

13 Major Characteristics of Each Measure of Central Tendency (2) Median It is the central value; 50% of the measurements lie above it and 50% fall below it. There is only one median for a data set. It is not influenced by extreme measurements. Medians of subsets cannot be combined to determine the median of the complete data set. For grouped data, its value is rather stable even when the data are organized into different categories. It is applicable to quantitative data only.

14 Major Characteristics of Each Measure of Central Tendency (3) Mean It is arithmetic average of the measurements in a data set. There is only one mean for a data set. Its value is influenced by extreme measurements; trimming can help to reduce the degree of influence. Means of subsets can be combined to determine the mean of the complete data set. It is applicable to quantitative data only.

15 Measures of Variability (1) Variability The description of the dispersion or spread of the measurements. Relative Frequency Relative Frequency Relative Frequency y y y

16 Measures of Variability (2) Range It is defined to be the difference between the largest and the smallest measurements of the data set. The simplest but least useful measure of data variability is the range. For grouped data, because the individual measurements are not known, the range is taken to be the difference between the upper limit of the last interval and the lower limit of the first interval.

17 Measures of Variability (3) Percentile The p-th percentile of a set of n measurements arranged in order of magnitude is that value that has at most p% of the measurements below it and at most (100-p)% above it. Specific percentiles of interest are the 25 th, 50 th, and 75 th percentiles, often called the lower quartile, the middle quartile(median), and the upper quartile, respectively. Relative Frequency % 25% 25% 25% Lower quartile Median IQR Upper quartile

18 Measures of Variability (4) For grouped data, the following formula can be used to approximate the percentiles for the original data. percentile of = cumulative frequency for all class intervals brfore the percentile class = frequency of w = interval width (( P /100) n cf P = percentile of interest L = lower limit of n = total frequency cf f p b P = L + w f p the class interval that includes interest b ) the class interval that includes the percentile of interest

19 Measures of Variability (5) Interquartile range (IQR) The difference between the upper and lower quartiles IQR = 75 th percentile 25 th percentile IQR ignores the extremes in the data set completely. The IQR does not provide a lot of useful information about the variability of a single set of measurements, but can be quite useful when comparing the variabilities of two or more data sets.

20 Measures of Variability (6) Variance Many different measures of variability can be constructed by using the deviation y y. The measure which involves the sum of the squared deviation of the measurements form their mean is called the variance. Population variance: n 2 ( y y) 2 i= 1 σ = n Sample variance: s 2 = n i= 1 ( y n 1 The use of (n-1) as the denominator of s 2 is not arbitrary. This definition of the sample variance makes it an unbiased estimator of the population variance σ 2. y) 2

21 Measures of Variability (7) Standard deviation The positive square root of the variance One reason for defining the standard deviation is that it yields a measure of variability having the same units of measurement as the original data, whereas the units for variance are the square of the measurement units. The other reason of using standard deviation is to apply the empirical rule to mound-shaped or bell-shaped distribution. Empirical rule Given a set of n measurements possessing a mound-shaped distribution, then the interval y ± s contains about 68% of the measurements; then the interval y ± 2s contains about 95% of the measurements; then the interval y ± 3s contains about 99.7% of the measurements;

22 Measures of Variability (8) Approximating s The empirical rule of standard deviation states that approximately 95% of the measurements lie in the interval y ± 2s. The length of this interval is, therefore, 4s. Because the range of the measurements is approximately 4s, we can obtain an approximate value for s by dividing the range by 4. approximat e value = range 4

23 Measures of Variability (8) Coefficient of variation (CV) It measures the variability in the values in a population relative to the magnitude of the population mean. CV = σ µ The CV is a unit-free number because the standard deviation and the mean are measured using the same units. It can be used to compare the variability in two considerably different processes or populations. In many applications, the CV is expressed as a percentage: CV σ =100 ( )% µ

24 Examination of the Shape of a Distribution Stem-and leaf plot Boxplot The boxplot is more concerned with the symmetry of the distribution and incorporates numerical measures of central tendency and location to study the variability of the measurements and the measurements in the tails of the distribution. Box-and-whiskers plot Any value beyond an inner fence on either side is called a mild outlier, and a value beyond an outer fence on either side is called an extreme outlier.

25 Homework 3.42 (p.94) 3.44 (p.94) 3.65 (p.112) 3.72, 73, 74 (p.113)

26 Pie Chart 63.6% Cola 8.6% Other 5.8% Orange 3.0% Root beer 12.7% Lemon-lime 6.3% Dr Pepper-type

27 Bar Chart Number of Workers Great Britain West Germany Japan Netherlands Ireland City

28 Frequency Histogram 12 Frequency Class interval for weight gain

29 Relative Frequency Histogram 0.12 Relative frequency Class intervals for weight gain

30 Influence of the Number of Class Intervals on the Appearance of Histograms Frequency Frequency Frequency Chick weight gain(grams) Chick weight gain(grams) Chick weight gain(grams)

31 Uniform Distribution Frequency Y

32 Normal Distribution Y Frequency

33 Unimodal Distribution (right-skewed) Frequency Y

34 Unimodal Distribution (left-skewed) Frequency Y

35 Bimodal Distribution Y Frequency

36 Bimodal Distribution (left-skewed) Frequency Y

37 Stem-and-leaf Plot N = 200 Median = Quartiles = , Decimal point is at the colon 1 : : : : : : : : : : : : : : 57 8 : 24 8 : : 02 9 : 9 High: High:

38 Example 1. Median Each of 10 children in the second grade was given a reading aptitude test. The scores were as follows: id grade Determine the median test score. Solution: Sort these scores: Because there are an event number of measurements, the median is the average of the two midpoint scores. median = = 85

39 Example 2. Grouped Data Median Class Interval f i cum(f i ) f i /n cum(f i /n) Totals n = The frequency table for the chick data. Compute the median weight gain for these data. L = f m 4.25 = 11 n = 100 w = 0.1 cf b = 47 median = L + w f m = (0.5n cf ) 0.1 (50 47) = 11 b 4.28

40 Example 3. Grouped Data Mean Class Interval f i y i f i y i f i (y i -y) Totals n = The actual value of the sample mean is: y = n i i= 1 = = n y To use the grouped data formula to calculate the mean: y n f i i i= 1 = = n y 4.292

41 Relation Among Mean, Trimmed Mean, Median and Mode (1) µ TM M d M o

42 Relation Among Mean, Trimmed Mean, Median and Mode (2) M o M d TM µ

43 Relation Among Mean, Trimmed Mean, Median and Mode (3) µ TM Md M o

44 Boxplot (1) Upper inner fence: Q (IQR) Q 3 Median Q 1 Upper outer fence: Q 3 +3(IQR) Lower outer fence: Q 1-3(IQR) Lower inner fence: Q 1-1.5(IQR)

45 Boxplot (2)

46 Boxplot (3)

47 Scatterplot age base