CSC Advanced Scientific Programming, Spring Descriptive Statistics

Transcription

1 CSC Advanced Scientific Programming, Spring 2018 Descriptive Statistics

2 Overview Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions. Data consists of information coming from observations, counts, measurements, or responses. A population is the collection of all outcomes, responses, measurements, or counts that are of interest. A sample is a subset of the population. A parameter is a numerical description of a population characteristic. A statistic is a numerical description of a sample characteristic.

3 Branches of Statistics Descriptive statistics is the branch of statistics that involves the organization, summarization, and display of data. Inferential statistics is the branch of statistics that involves using a sample to draw conclusions about a population. A basic tool in the study of inferential statistics is probability.

4 Data Classification Types of data: Qualitative data consist of attributes, labels, or nonnumerical entries. Quantitative data consist of numerical measurements or counts. Levels of measurement: Nominal: categorized using names, labels, or qualities. Ordinal: can be arranged in order or ranked. Interval: can be ordered and meaningful differences between entries can be calculated. Ratio: similar to interval, but there is a zero entry that is an inherent zero (implies none).

5 Frequency Distributions A frequency distribution is a table that shows classes or intervals of data entries with a count of the number of entries in each class. The frequency f of a class is the number data entries in the class. Constructing a frequency distribution: 1 Decide on the number of classes. 2 Find the class width: the range of the data divided by the number of classes rounded up to a convenient number. 3 Find the class limits 4 Count up the number of data entries that fall within the class boundaries to determine the frequency f for each class.

6 Frequency Distribution Example Number of classes: 5 Data set: 7, 39, 13, 9, 25, 8, 22, 0, 2, 18, 2, 30, 7, 35, 12, 15, 8, 6, 5, 29, 0, 11, 39, 16, 15 Range = 39-0 Class width = 39 5 = 7.8 = 8 Frequency distribution: Class f f = 25

7 Frequency Distributions The midpoint of a class is the sum of the lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark. The relative frequency of a class is the portion or percentage of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n. The cumulative frequency of a class is the sum of the frequency for that class and all previous classes. The cumulative frequency of the last class is equal to the sample size n.

8 Frequency Distribution Example Number of classes: 5 Data set: 7, 39, 13, 9, 25, 8, 22, 0, 2, 18, 2, 30, 7, 35, 12, 15, 8, 6, 5, 29, 0, 11, 39, 16, 15 Frequency distribution: Class f Midpoint Relative Cumulative f = 25 f n = 1

9 Measures of Central Tendency The mean of a data set is the sum of the data entries divided by the number of entries. Population mean: x µ = N Sample mean: x x = n The median of a data set is the value that lies in the middle of the data when the data is in sorted order. The mode of a data set is the data entry that occurs with the greatest frequency.

10 Measures of Central Tendency An outlier is a data entry that is far removed from the other entries in the data set. A weighted mean is the mean of a data set whose entries have varying weights. A weighted mean is given by: x = x w w where w is the weight of each entry x.

11 Measures of Central Tendency The mean of a frequency distribution for a sample is approximated by x f x = f where f is the frequency and x is the midpoint.

12 Frequency Distribution Example Number of classes: 5 Data set: 7, 39, 13, 9, 25, 8, 22, 0, 2, 18, 2, 30, 7, 35, 12, 15, 8, 6, 5, 29, 0, 11, 39, 16, 15 Frequency distribution: Class f Midpoint, x x f f = 25 x f = Mean of frequency distribution: = 14.7

13 Measures of Variation The range of a data set is the difference between the maximum and minimum data entries in the set. The deviation of an entry x in a population data set is the difference between the entry and the mean µ of the data set. Deviation of x = x µ The population variance of a population data set of N entries is (x µ) Population variance = σ 2 2 = N where the symbol σ is a lowercase Greek letter Sigma.

14 Measures of Variation The population standard deviation of a population data set of N entries is the square root of the population variance σ = (x µ) σ 2 2 = N

15 Finding Population Variance and Standard Deviation 1. Find the mean of the population data set. µ = 2. Find the devation of each entry. x µ x N 3. Square each deviation. (x µ) 2 4. Add to get the sum of squares SS x = (x µ) 2 5. Divide by N to get the population variance. σ 2 = 6. Find the square root of the variance to get the population standard deviation. σ = (x µ) 2 N (x µ) 2 N

16 Measures of Variation The sample variance and sample standard deviation of a sample data set of n entries are Sample variance = s 2 = (x x) 2 n 1 (x x) 2 Sample standard deviation = s = n 1

17 Measures of Variation Symbols Population Sample Variance σ 2 s 2 Standard deviation σ s Mean µ x Number of entries N n Deviation x µ x x Sum of squares (x µ) 2 (x x) 2

18 Measures of Variation The standard deviation of a frequency distribution is: (x x) 2 f s = n 1 where n = f.