Fundamentals of Statistics EBB 341

Statistics? A collection of quantitative data from a sample or population. The science that deals with the collection, tabulation, analysis, interpretation, and presentation of quantitative data.

Statistic types Deductive or descriptive statistics describe and analyze a complete data set Inductive statistics deal with a limited amount of data (sample). Conclusions: probability?

Population A population is any entire collection of people, animals, plants or things from which we may collect data. It is the entire group we are interested in, which we wish to describe or draw conclusions about. For each population there are many possible samples.

Sample A sample is a group of units selected from a larger group (population). By studying the sample it is hoped to draw valid conclusions about population. The sample should be representative of the general population. The best way is by random sampling.

Parameter A parameter is a value, usually unknown (and which therefore has to be estimated), used to represent a certain population characteristic. For example, the population mean is a parameter that is often used to indicate the average value of a quantity.

Statistics Parameters: 2 POPULATION Inferential Statistics Deductive SAMPLE Statistics: x, s, s2 Inductive

Inferential Statistics Statistical Inference makes use of information from a sample to draw conclusions (inferences) about the population from which the sample was taken.

Types of data Variables data Attribute data quality characteristics that are measurable values. measurable and normally continuous; may take on any value - eg. weight in kg Attribute data quality characteristics that are observed to be either present or absent, conforming or nonconforming. countable and normally discrete; integer - eg: 0, 1, 5, 25, …, but cannot 4.65

Accurate and Precise Data life of light bulb: 995.6 h The value of 995.632 h, is too accurate & unnecessary Keyway spec: lower limit 9.52 mm, upper limit 9.58 mm – data collected to the nearest 0.001 mm, and rounded to nearest 0.01 mm.

Accurate and Precise Measuring instruments may not give a true reading because of problems due to accuracy and precision. Data: 0.9532, 0.9534 = 0.953 Data: 0.9535, 0.9537 = 0.954 If the last digit is 5 or greater, rounded up

Describing the Data Graphical: Analytical: Plot or picture of a frequency distribution. Analytical: Summarize data by computing a measure of central tendensy and dispersion.

Sampling Methods Sampling methods are methods for selecting a sample from the population: Simple random sampling - equal chance for each member of the population to be selected for the sample. Systematic sampling - the process of selecting every n-th member of the population arranged in a list. Stratified sample - obtained by dividing the population into subgroups and then randomly selecting from each subgroups. Cluster sampling - In cluster sampling groups are selected rather than individuals. Incidental or convenience sampling - Incidental or convenience sampling is taking an intact group (e.g. your own forth grade class of pupils)

Frequency Distribution Data Set - High Temperatures for 30 Days 50 45 49 43 47 44 51 46 Consider the following set of data which are the high temperatures recorded for 30 consequetive days. We wish to summarize this data by creating a frequency distribution of the temperatures.

To create a frequency distribution Identify the highest and lowest values (51 & 43). Create a column with variable, in this case temp. Enter the highest score at the top, and include all values within the range from highest score to lowest score. Create a tally column to keep track of the scores. Create a frequency column. At the bottom of the frequency column record the total frequency.

To create a frequency distribution Frequency Distribution for High Temperatures Temperature Tally Frequency 51 //// 4 50 49 ////// 6 48 47 /// 3 46 45 44 43   N = 30 Identify the highest and lowest values (51 & 43). Create a column with variable, in this case temp. Enter the highest score at the top, and include all values within the range from highest score to lowest score. Create a tally column to keep track of the scores. Create a frequency column. At the bottom of the frequency column record the total frequency.

Frequency Distribution for High Temperatures Tally Frequency 51 //// 4 50 49 ////// 6 48 47 /// 3 46 45 44 43   N = 30 Frequency Distribution

Cummulative Frequency Distribution A cummulative freq distribution can be created by adding an additional column called "Cummulative Frequency." The cum. frequency for a given value can be obtained by adding the frequency for the value to the cummulative value for the value below the given value. For example: The cum. frequency for 45 is 10 which is the cum. frequency for 44 (6) plus the frequency for 45 (4). Finally, notice that the cum. frequency for the highest value should be the same as the total of the frequency column.

Cummulative Frequency Distribution for High Temperatures Tally Frequency Cummulative Frequency 51 //// 4 30 50 26 49 ////// 6 22 48   16 47 /// 3 46 13 45 10 44 43 N =

Grouped frequency distribution In some cases it is necessary to group the values of the data to summarize the data properly. Eg., we wish to create a freq. distribution for the IQ scores of 30 pupils. The IQ scores in the range 73 to 139. To include these scores in a freq. distribution we would need 67 different score values (139 down to 73). This would not summarize the data very much. To solve this problem we would group scores together and create a grouped freq. distribution. If data has more than 20 score values, we should create a grouped freq. distribution by grouping score values together into class intervals.

To create a grouped frequency distribution: select an interval size (7-20 class intervals) create a class interval column and list each of the class intervals each interval must be the same size, they must not overlap, there may be no gaps within the range of class intervals create a tally column (optional) create a midpoint column for interval midpoints create a frequency column enter N = sum value at the bottom of the frequency column

Data Set - High Temperatures for 50 Days Grouped frequency Data Set - High Temperatures for 50 Days 57 39 52 43 50 53 42 58 55 49 45 51 44 54 59 41 40 47 46 Look at the following data of high temperatures for 50 days. The highest temperature is 59 and the lowest temperature is 39. We would have 21 temperature values. This is greater than 20 values so we should create a grouped frequency distribution.

Grouped Frequency Distribution for High Temperatures Class Interval Tally Interval Midpoint Frequency 57-59 ////// 58 6 54-56 /////// 55 7 51-53 /////////// 52 11 48-50 ///////// 49 9 45-47 46 42-44 43 39-41 //// 40 4   N = 50

Cumulative grouped frequency distribution Cumulative Grouped Frequency Distribution for High Temperatures Class Interval Tally Interval Midpoint Frequency Cumulative Frequency 57-59 //// / 58 6 50 54-56 //// // 55 7 44 51-53 //// //// / 52 11 37 48-50 //// //// 49 9 26 45-47 46 17 42-44 43 10 39-41 //// 40 4   N =

To create a histogram from this frequency distribution Arrange the values along the abscissa (horizonal axis) of the graph Create a ordinate (vertical axis) that is approximately three fourths the length of the abscissa, to contain the range of scores for the frequencies. Create the body of the histogram by drawing a bar or column, the length of which represents the frequency for each age value. Provide a title for the histogram.

High temperatures for 50 days Frequency Temperatures

Histograms Constructing a Histogram for Discrete Data First, determine the frequency and relative frequency of each x value. Then mark possible x value on a horizontal scale.

Cara Menyediakan Histogram -Grouped Data Tentukan nilai perbezaan, R = nilai terbesar – nilai terkecil atau R = Xh - Xl Dapatkan bilangan turus histogram, Kira lebar turus, h = R/t Nilai permulaan turus = nilai terkecil data – (h/2) atau Xl – (h/2) Lukis histogram.

Histograms Constructing a Histogram for Continuous Data: equal class width Number of classes  Data Relative frequency

Bar Graph A bar graph is similar to a histogram except that the bars or columns are seperated from one another by a space rather than being contingent to one another. The bar graph is used to represent categorical or discrete data, that is data at the nominal or ordinal level of measurement. The variable levels are not continuous.

Bar Graph 11 Ed Admin 3 1 2 5 N = 24 Frequency Major Counseling Elem Educ 1 Music Educ Reading 2 Social Work Special Educ 5 N = 24

Descriptive statistics Measures of Central Tendency Describes the center position of the data Mean, Median, Mode Measures of Dispersion Describes the spread of the data Range, Variance, Standard deviation

Measures of central tendency: Mean Arithmetic mean: x = where xi is one observation,  means “add up what follows” and N is the number of observations So, for example, if the data are : 0,2,5,9,12 the mean is (0+2+5+9+12)/5 = 28/5 = 5.6

Frequency Distribution of Ages for Children in After School Program Mean for a Population Science Test Scores 17 23 27 26 25 30 19 24 29 18 22 21 Frequency Distribution of Ages for Children in After School Program Age Frequency fX 11 2 22 10 4 40 9 8 72 7 56 3 21 6 5 1 N = 25 216

Mean for a Sample Ungrouped data: Grouped data: n= number of observed values n = sum of the frequencies h= number of cells or number observed values Xi = cell midpoint

Example: - ungrouped data Resistance of 5 coils: 3.35, 3.37, 3.28, 3.34, 3.30 ohm. The average:

Example: - grouped data Frequency Distributions of the life of 320 tires in 1000 km Boundaries Midpoint, Xi Frequency, fi Computation, fiXi 23.6-26.5 25.0 4 100 26.6-29.5 28.0 36 1008 29.6-32.5 31.0 51 1581 32.6-35.5 34.0 63 2142 35.6-38.5 37.0 58 2146 38.6-41.5 40.0 52 2080 41.6-44.5 43.0 34 1462 44.6-47.5 46.0 16 736 47.6-50.5 49.0 6 294 Total n = 320  fiXi = 11549

Measures of Location Central tendency Data: sample mean sample median Provided that data is in increasing order e.g. data: 2, 2, 3, 4, 15 Median is less sensitive to outliers.

Median - mode Median = the observation in the ‘middle’ of sorted data Mode = the most frequently occurring value

Median and mode 100 91 85 84 75 72 72 69 65 Mode Median Mean = 79.22

Median Grouped data: Lm= lower boundary with the median cfm= cumulative freq. all cells below Lm fm= freq. median i = cell interval

Median - Grouped technique Use data from table above (Frequency Distributions of the life of 320 tires in 1000 km). The halfway point (320/2 = 160) is reached in the cell with midpoint value of 37.0 and a lower limit of 35.6. The cumulative frequency is 4+36+51+63 is 154, the cell interval is 3, and the frequency of the median cell is 58: Median = 35.9 x 1000 km = 35900 km.

Measures of dispersion: range The range is calculated by taking the maximum value and subtracting the minimum value. 2 4 6 8 10 12 14 Range = 14 - 2 = 12

Measures of dispersion: variance Calculate the deviation from the mean for every observation. Square each deviation Add them up and divide by the number of observations

Variance for a population Worksheet for Calculating the Variance for 7 scores 5 1 3 -1 4 28 Variance for a population The formula for the variance for a population using the deviation score method is as follows: The mean = 28/7 = 4 The population variance:

Measures of dispersion: standard deviation The standard deviation is the square root of the variance. The variance is in “square units” so the standard deviation is in the same units as x.

Standard Deviation for a Sample General formula/ungrouped data: For computation purposes:

Standard Deviation for a Sample Grouped data:

Example- ungrouped data Sample: Moisture content of kraft paper are 6.7, 6.0, 6.4, 6.4, 5.9, and 5.8 %. Sample standard deviation, s = 0.35 %

Calculating the Sample Standard Deviation - Grouped technique Standard deviation for a grouped sample: Average: Table: Car speeds in km/h Boundaries Xi fi fiXi fiXi2 72.6-81.5 77.0 5 385 29645 81.6-90.5 86.0 19 1634 140524 90.6-99.5 95 31 2945 279775 99.6-108.5 104.0 27 2808 292032 108.6-117.5 113 14 1582 178766 Total 96 9354 920742

Skewness a3 = 0, symmetrical a3 > 0 (positive), the data are skewed to the right, means that long the long tail is to right a3 < 0 (negative), skewed to the left, means that long the long tail is to left

Kurtosis Leptokurtic (more peaked) distribution Platykurtic (flatter) distribution Mesokurtic (between these 2 distribution – normal distribution. For example, if a normal distribution, mesokurtic, has a4 = 3, a4 > 3 is more peaked than normal a4 < 3 is less peaked than normal.

Example: That data are skewed to the left Xi fi Xi - fi (Xi- )3 1 4 (1-7) = -6 -864 5184 24 (4-7) = -3 -648 1944 7 64 (7-7) = 0 10 32 (10-7) = +3 +864 2592   124 9720 That data are skewed to the left

Standard deviation and curve shape If  is small, there is a high probability for getting a value close to the mean. If  is large, there is a correspondingly higher probability for getting values further away from the mean.

The Normal Curve The normal curve or the normal frequency distribution or Gaussian distribution is a hypothetical distribution that is widely used in statistical analysis. The characteristics of the normal curve make it useful in education and in the physical and social sciences.

Characteristics of the Normal Curve The normal curve is a symmetrical distribution of data with an equal number of data above and below the midpoint of the abscissa. Since the distribution of data is symmetrical the mean, median, and mode are all at the same point on the abscissa. In other words, mean = median = mode. If we divide the distribution up into standard deviation units, a known proportion of data lies within each portion of the curve.

34.13% of data lie between  and 1 above the mean (). 34.13% between  and 1 below the mean. Approximately two-thirds (68.28 %) within 1 of the mean. 13.59% of the data lie between one and two standard deviations Finally, almost all of the data (99.74%) are within 3 of the mean.

The normal curve If x follows a bell-shaped (normal) distribution, then the probability that x is within 1 standard deviation of the mean is 68% 2 standard deviations of the mean is 95 % 3 standard deviations of the mean is 99.7%

Standardized normal value, Z When a score is expressed in standard deviation units, it is referred to as a Z-score. A score that is one standard deviation above the mean has a Z-score of 1. A score that is one standard deviation below the mean has a Z-score of -1. A score that is at the mean would have a Z-score of 0. The normal curve with Z-scores along the abscissa looks exactly like the normal curve with standard deviation units along the abscissa.

Z-value Deviation IQ Scores, sometimes called Wechsler IQ scores, are a standard score with a mean of 100 and a standard deviation of 15. What percentage of the general population have deviation IQs lower than 85? So an IQ of 85 is equivalent to a z-value of –1. So 50 % - 34.13 % = 15.87% of the population has IQ scores lower than 85.

Frequency Polygon A frequency polygon is what you may think of as a curve. A frequency polygon can be created with interval or ratio data. Let's create a frequency polygon with the data we used earlier to create a histogram.

To create a frequency polygon Arrange the values along the abscissa (horizonal axis). Arrange the lowest data on the left & the highest on the right. Add one value below the lowest data and one above the highest data. Create a ordinate (vertical axis). Arrange the frequency values along the abscissa. Provide a label for the ordinate (Frequency). Create the body of the frequency polygon by placing a dot for each value. Connect each of the dots to the next dot with a straight line. Provide a title for the frequency polygon.

To create a frequency polygon