CHAPTER 3  Descriptive Statistics Measures of Central Tendency 1

Descriptive Statistics Measures of Central Tendency  Mean  Interval or Ratio scale  Polygon ean –The sum of the values divided by the number of values--often called the "average." μ=ΣX/N –Add all of the values together. Divide by the number of values to obtain the mean. –Example: X ???? ???? 2

Descriptive Statistics The Mean is: μ=ΣX/N= 82/5=16.4 ( ) / 5 =

The Characteristics of Mean  1. Changing a score in a distribution will change the mean  2. Introducing or removing a score from the distribution will change the mean  3. Adding or subtracting a constant from each score will change the mean  4. Multiplying or dividing each score by a constant will change the mean  5. Adding a score which is same as the mean will not change the mean 4

Descriptive Statistics  Measures of Central Tendency  Median/Middle  Ordinal Scale  Bar/Histogram edian –Divides the values into two equal halves, with half of the values being lower than the median and half higher than the median.  Sort the values into ascending order.  If you have an odd number of values, the median is the middle value.  If you have an even number of values, the median is the arithmetic mean (see above) of the two middle values. –Example: The median of the same five numbers (7, 12, 24, 20, 19) is ???. 5

Statistics  The median is 19.  Mode-  Nominal Scale  Bar/Histogram ode –The most frequently-occurring value (or values).  Calculate the frequencies for all of the values in the data.  The mode is the value (or values) with the highest frequency. –Example: For individuals having the following ages -- 18, 18, 19, 20, 20, 20, 21, and 23, the mode is ???? The Mode is 20 6

CHARACTERISTICS OF MODE  Nominal Scale  Discrete Variable  Describing Shape 7

The Range  The Range: The Range is the difference between the highest number –lowest number +1 The Range is the difference between the highest number –lowest number +1 2, 4, 7, 8, and 10 -> Discrete Numbers 2, 4, 7, 8, and 10 -> Discrete Numbers 2, 4.6, 7.3, 8.4, and 10 -> Continues Numbers 2, 4.6, 7.3, 8.4, and 10 -> Continues Numbers The difference between the upper real limit of the highest number and the lower real limit of the lowest number. The difference between the upper real limit of the highest number and the lower real limit of the lowest number.

CHAPTER 4  Variability 9

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Variability  Variability is a measure of dispersion or spreading of scores around the mean, and has 2 purposes:  1. Describes the distribution Next slide Next slide 11

Range, Interquartile Range, Semi-Interquartile Range, Standard Deviation, and Variance are the Measures of Variability  The Range: The Range is the difference between the highest number –lowest number +1 The Range is the difference between the highest number –lowest number +1 2, 4, 7, 8, and 10 -> Discrete Numbers 2, 4, 7, 8, and 10 -> Discrete Numbers 2, 4.6, 7.3, 8.4, and 10 -> Continues Numbers 2, 4.6, 7.3, 8.4, and 10 -> Continues Numbers The difference between the upper real limit of the highest number and the lower real limit of the lowest number. The difference between the upper real limit of the highest number and the lower real limit of the lowest number.

Variability  2. How well an individual score (or group of scores) represents the entire distribution. i.e. Z Score  Ex. In inferential statistics we collect information from a small sample then, generalize the results obtained from the sample to the entire population. 13

Interquartile Range (IQR) Interquartile Range (IQR)  In descriptive statistics, the Interquartile Range (IQR), also called the midspread or middle fifty, is a measure of statistical dispersion, being equal to the difference between the upper and lower quartiles. (Q 3 − Q 1 )=IQR descriptive statistics statistical dispersion quartilesdescriptive statistics statistical dispersion quartiles 14

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Interquartile Range (IQR)  IQR is the range covered by the middle 50% of the distribution.  IQR is the distance between the 3 rd Quartile and 1 st Quartile. 18

Semi-Interquartile Range (SIQR)  SIQR is ½ or half of the Interquartile Range.  SIQR = (Q3-Q1)/2 19

Variability 20

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Variability Range, SS, Standard Deviations and Variances  X σ² = ss/N Pop 1 σ = √ss/N 1 σ = √ss/N 2 4 s² = ss/n-1 or ss/df Standard deviation 4 s² = ss/n-1 or ss/df Standard deviation 5 s = √ss/df Sample 5 s = √ss/df Sample SS= Σx ²-(Σx)²/N  Computation SS= Σ ( x- μ ) ²  Definition Sum of Squared Deviation from Mean Variance (σ²) is the Mean of Squared Deviations=MS 22

23 Practical Implication for Test Construction Variance and Covariance measure the quality of each item in a test. Reliability and validity measure the quality of the entire test.  σ²=SS/N  used for one set of data Variance is the degree of variability of scores from mean. Correlation is based on a statistic called Covariance (Cov xy or S xy) ….. r=sp/√ssx.ssy  COVxy=SP/N-1  used for 2 sets of data Covariance is a number that reflects the degree to which 2 variables vary together.

24 Variance  X σ² = ss/N Pop 1 s² = ss/n-1 or ss/df Sample 1 s² = ss/n-1 or ss/df Sample SS= Σx ²-(Σx)²/N SS= Σ ( x- μ ) ² Sum of Squared Deviation from Mean

25Covariance  Correlation is based on a statistic called Covariance (Cov xy or S xy ) ….. COV xy =SP/N-1 Correlation--  r=sp/√ssx.ssy Correlation--  r=sp/√ssx.ssy  Covariance is a number that reflects the degree to which 2 variables vary together.  Original Data X Y X Y

26Covariance  Correlation is based on a statistic called Covariance (Cov xy or S xy ) ….. COV xy =SP/N-1 Correlation--  r=sp/√ssx.ssy Correlation--  r=sp/√ssx.ssy  Covariance is a number that reflects the degree to which 2 variables vary together.  Original Data X Y X Y

 27Covariance

28 Descriptive Statistics for Nondichotomous Variables

29 Descriptive Statistics for Dichotomous Data

30 Descriptive Statistics for Dichotomous Data Item Variance & Covariance

FACTORS THAT AFFECT VARIABILITY  1. Extreme Scores i.e. 1, 3, 8, 11, 1,000, We can’t use the Range in this situation but we can use the other measures of variability.  2. Sample Size If we increase the sample size will change the Range therefore we can’t use the Range in this situation but we can use the other measures of variability.  3. Stability Under Sampling (see next slide) p.130 The S and S² for all samples should be the same because they come from same population (all slices of a pizza should taste the same).  4. Open-Ended Distribution When we don’t have highest score and lowest score in a distribution 31

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