Descriptive Statistics Ernesto Diaz Faculty – Mathematics Redwood High School 1

Basic Concepts In statistics a population, includes all of the items of interest, and a sample, includes some of the items in the population. The study of statistics can be divided into two main areas. Descriptive statistics, has to do with collecting, organizing, summarizing, and presenting data (information). Inferential statistics, has to do with drawing inferences or conclusions about populations based on information from samples.

Basic Concepts Information that has been collected but not yet organized or processed is called raw data. It is often quantitative (or numerical), but can also be qualitative (or nonnumerical).

Basic Concepts Quantitative data: The number of siblings in ten different families: 3, 1, 2, 1, 5, 4, 3, 3, 8, 2 Qualitative data: The makes of five different automobiles: Toyota, Ford, Nissan, Chevrolet, Honda Quantitative data can be sorted in mathematical order. The number siblings can appear as 1, 1, 2, 2, 3, 3, 3, 4, 5, 8

Measures of Central Tendency Mean Median Mode Symmetry in Data Sets © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Mean The mean (more properly called the arithmetic mean) of a set of data items is found by adding up all the items and then dividing the sum by the number of items. (The mean is what most people associate with the word “average.”) The mean of a sample is denoted (read “x bar”), while the mean of a complete population is denoted (the lower case Greek letter mu). © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Mean The mean of n data items x1, x2,…, xn, is given by the formula We use the symbol for “summation,” (the Greek letter sigma). © 2008 Pearson Addison-Wesley. All rights reserved

Example-find the mean Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows: $327 $465 $672 $150 $230

© 2008 Pearson Addison-Wesley. All rights reserved Weighted Mean The weighted mean of n numbers x1, x2,…, xn, that are weighted by the respective factors f1, f2,…, fn is given by the formula © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Median Another measure of central tendency, which is not so sensitive to extreme values, is the median. This measure divides a group of numbers into two parts, with half the numbers below the median and half above it. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Median To find the median of a group of items: Step 1 Rank the items. Step2 If the number of items is odd, the median is the middle item in the list. Step 3 If the number of items is even, the median is the mean of the two middle numbers. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Median Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 1, 6, 3, 3, 4, 2. Find the median number of siblings for the ten students. Solution In order: 1, 1, 2, 2, 2, 3, 3, 3, 4, 6 Median = (2+3)/2 = 2.5 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Mode The mode of a data set is the value that occurs the most often. Sometimes, a distribution is bimodal (literally, “two modes”). In a large distribution, this term is commonly applied even when the two modes do not have exactly the same frequency © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Mode for a Set Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mode for the number of siblings. Solution 3, 2, 2, 1, 3, 6, 3, 3, 4, 2 The mode for the number of siblings is 3. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Mode for Distribution Find the median for the distribution. Value 1 2 3 4 5 Frequency 6 8 Solution The mode is 5 since it has the highest frequency (8). © 2008 Pearson Addison-Wesley. All rights reserved

Measures of Position Measures of position are often used to make comparisons. Two measures of position are percentiles and quartiles.

To Find the Quartiles of a Set of Data Order the data from smallest to largest. Find the median, or 2nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data.

To Find the Quartiles of a Set of Data continued The first quartile, Q1, is the median of the lower half of the data; that is, Q1, is the median of the data less than Q2. The third quartile, Q3, is the median of the upper half of the data; that is, Q3 is the median of the data greater than Q2.

Example: Quartiles The weekly grocery bills for 23 families are as follows. Determine Q1, Q2, and Q3. 170 210 270 270 280 330 80 170 240 270 225 225 215 310 50 75 160 130 74 81 95 172 190

Example: Quartiles continued Order the data: 50 75 74 80 81 95 130 160 170 170 172 190 210 215 225 225 240 270 270 270 280 310 330 Q2 is the median of the entire data set which is 190. Q1 is the median of the numbers from 50 to 172 which is 95. Q3 is the median of the numbers from 210 to 330 which is 270.

Measures of Dispersion

Measures of Dispersion Sometimes we want to look at a measure of dispersion, or spread, of data. Two of the most common measures of dispersion are the range and the standard deviation.

Measures of Dispersion Measures of dispersion are used to indicate the spread of the data. The range is the difference between the highest and lowest values; it indicates the total spread of the data.

Example: Range Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries. $24,000 $32,000 $26,500 $56,000 $48,000 $27,000 $28,500 $34,500 $56,750 Range = $56,750  $24,000 = $32,750

Standard Deviation One of the most useful measures of dispersion, the standard deviation, is based on deviations from the mean of the data. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Deviations from the Mean Find the deviations from the mean for all data values of the sample 1, 2, 8, 11, 13. Solution The mean is 7. Subtract to find deviation. Data Value 1 2 8 11 13 Deviation –6 –5 4 6 13 – 7 = 6 The sum of the deviations for a set is always 0. © 2008 Pearson Addison-Wesley. All rights reserved

Standard Deviation The variance is found by summing the squares of the deviations and dividing that sum by n – 1 (since it is a sample instead of a population). The square root of the variance gives a kind of average of the deviations from the mean, which is called a sample standard deviation. It is denoted by the letter s. (The standard deviation of a population is denoted the lowercase Greek letter sigma.) © 2008 Pearson Addison-Wesley. All rights reserved

Standard Deviation The standard deviation measures how much the data differ from the mean.

Calculation of Standard Deviation The individual steps involved in this calculation are as follows Step 1 Calculate the mean of the numbers. Step 2 Find the deviations from the mean. Step 3 Square each deviation. Step 4 Sum the squared deviations. Step 5 Divide the sum in Step 4 by n – 1. Step 6 Take the square root of the quotient in Step 5. © 2008 Pearson Addison-Wesley. All rights reserved

Example Solution Data Value 1 2 8 11 13 Deviation –6 –5 4 6 Find the standard deviation of the sample 1, 2, 8, 11, 13. Solution The mean is 7. Data Value 1 2 8 11 13 Deviation –6 –5 4 6 (Deviation)2 36 25 16 Sum = 36 + 25 + 1 + 16 + 36 = 114 © 2008 Pearson Addison-Wesley. All rights reserved

Example Solution (continued) Divide by n – 1 with n = 6: Take the square root: © 2008 Pearson Addison-Wesley. All rights reserved

Example $280, $217, $665, $684, $939, $299 Find the mean. Find the standard deviation of the following prices of selected washing machines: $280, $217, $665, $684, $939, $299 Find the mean.

Example continued, mean = 514 421,516 180,625 425 939 28,900 170 684 22,801 151 665 46,225 -215 299 54,756 -234 280 (-297)2 = 88,209 -297 217 (Data - Mean)2 Data - Mean Data

Example continued, mean = 514 The standard deviation is $290.35.

Interpreting Measures of Dispersion A main use of dispersion is to compare the amounts of spread in two (or more) data sets. A common technique in inferential statistics is to draw comparisons between populations by analyzing samples that come from those populations.

Example: Interpreting Measures Two companies, A and B, sell small packs of sugar for coffee. The mean and standard deviation for samples from each company are given below. Which company consistently provides more sugar in their packs? Which company fills its packs more consistently? Company A Company B

Example: Interpreting Measures Solution We infer that Company A most likely provides more sugar than Company B (greater mean). We also infer that Company B is more consistent than Company A (smaller standard deviation).

© 2008 Pearson Addison-Wesley. All rights reserved Symmetry in Data Sets The most useful way to analyze a data set often depends on whether the distribution is symmetric or non-symmetric. In a “symmetric” distribution, as we move out from a central point, the pattern of frequencies is the same (or nearly so) to the left and right. In a “non-symmetric” distribution, the patterns to the left and right are different. © 2008 Pearson Addison-Wesley. All rights reserved

Some Symmetric Distributions © 2008 Pearson Addison-Wesley. All rights reserved

Non-symmetric Distributions A non-symmetric distribution with a tail extending out to the left, shaped like a J, is called skewed to the left. If the tail extends out to the right, the distribution is skewed to the right. © 2008 Pearson Addison-Wesley. All rights reserved

Some Non-symmetric Distributions © 2008 Pearson Addison-Wesley. All rights reserved

Chebyshev’s Theorem For any set of numbers, regardless of how they are distributed, the fraction of them that lie within k standard deviations of their mean (where k > 1) is at least © 2008 Pearson Addison-Wesley. All rights reserved

Example: Chebyshev’s Theorem What is the minimum percentage of the items in a data set which lie within 3 standard deviations of the mean? Solution With k = 3, we calculate © 2008 Pearson Addison-Wesley. All rights reserved