![DEVIATION OF THE MEAN?
∑ (x - X‾) = 0 [ you will always get 0]
[ when squared and summed, mean deviation scores yield positive numbers
that are useful in a variety of statistical analyses]
• The sum of squares (ss) is the sum of the squared deviations from the mean
of a distribution. The ss is fundamental to descriptive and inferential
statistics.
SS = ∑ (X - X‾)
2](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/descriptivestatistics-180327162655/85/Descriptive-statistics-24-320.jpg)
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Descriptive statistics
- 1. DESCRIPTIVE STATISTICS CENTRAL TENDENCY AND VARIABILITY DESMOND AYIM-ABOAGYE, PHD
- 2. IMPORTANCE OF CENTRAL TENDENCY • The goal of using a measure of central tendency is to identify a numerical value that is the most representative one within a distribution or set of data. • Central tendency and variability are inextricably linked– in order to make sense of one you must know the other.
- 3. CENTRAL TENDENCY • Measures of central tendency are descriptive statistics that identify the central location of a sample of data. The central tendency of a data set is the best single indicator describing the representative value (s) of any sample.
- 4. MEAN • The mean is by far the single most useful measure of central tendency available, and it can be used to analyze data from interval or ratio scales of measurement. • The study of means in psychology literature is extensive (ANOVA, Analysis of Variance).
- 5. MEAN DEFINITION • The mean is the arithmetic average of a set of scores. The mean is calculated by summing the set of scores and dividing by the N of scores. • CALCULATING THE MEAN FOR A SAMPLE OF DATA (FORMULA) • ¯X = Σ𝑥/N = • ¯X = 𝑥1 + 𝑥2 + 𝑥3 + … + 𝑥N/N • ∑ = Total (from Greek word sigma) • ¯X = Mean • 𝑥 = scores • N = Sample
- 6. MEAN OF A POPULATION • CALCULATING THE MEAN FOR A POPULATION (FORMULA) • 𝜇 = ∑x/N
- 7. Sample Statistic Pop Parameter Unbiased Estimate of Population Parameters Mean Variance Standard deviation Pearson correlation coef. ¯X 𝑠2 S r 𝜇 𝜎 2 𝜎 𝜌 - ᵔs 2 ᵔs - Table 4.1. Some Symbols for Sample Statistics, Population Parameters, and Unbiased Estimates of Population Parameters.
- 8. x f fx 8 7 6 5 4 3 2 1 0 3 5 6 8 10 4 2 5 1 24 35 36 40 40 12 4 5 0 N = 44 ∑fx = 196 ¯X = Σ𝑓𝑥/N ¯X = 196/44 ¯X = 4.4545 ≅ 4.45
- 9. THE MEDIAN • The Median can be used for calculations involving ordinal, interval, or ratio scale data. • The Median is a number or score that precisely divides a distribution of data in half. Fifty percent of a distribution’s observations will fall above the median and fifty percent will fall below it.
- 10. 26 32 21 12 15 11 27 16 18 21 19 28 10 13 31 These are 15 scores to consider: 10 11 12 13 15 16 18 19 21 21 26 27 28 31 32 To calculate the median, arrange the scores from the lowest to the highest: 19 because 7 scores appear on either side of this median score
- 11. FORMULA FOR EVEN NUMBERS Median score = N + 1 2 15 +1 / 2 = 8 10 11 12 13 15 16 18 19 21 21 26 27 28 31 32
- 12. When you have odd number of scores that splits the distribution into two halves. Formula 55 67 78 83 88 92 98 99 (83 + 88)/2 = 171/2 = 85.5
- 13. THE MODE • The third and final measure of central tendency is the mode. • The mode is the most frequently occurring observation within a distribution. • The mode of a distribution is the score or category that has the highest frequency.
- 14. 10 11 12 13 15 16 18 19 21 21 26 27 27 27 27 31 31 45 The mode is 27
- 15. Y X Y X Y X Normal distribution Positive skew Negative skew mdn Xm d n m o d e mode X m d n Figure 4. Relation of Measures of Central Tendency to the Shape of Distributions
- 16. VARIABILITY • Think in terms of variability: • A) Are scores clustered close together? • B) Right on top of one another?, • C) Spread very far apart? • Clustering, Spreading, Dispersion – synonym for the concept of variability
- 17. VARIABILITY DEFINITION • Variability refers to the degree to which sample or population observations differ or deviate from the distribution’s measures of central tendency. • The clustering, spread, or dispersion of data emphasize the relative amount of variability present in a distribution. • The shape of a distribution, its skewness or kurtosis, often characterizes its variability.
- 18. FACTORS AFFECTING VARIABILITY • 1. Sample Size • 2. Selection Process • 3. Sample Characteristics • 4. Independent Variables • 5. Dependent measures • 6. Passage of time between the presentation of the independent variable and dependent measure.
- 19. THE RANGE • The most basic index of variability is the range. It can be crude range or even the simple range. • The range is the difference between the highest and the lowest score in a distribution. Formula: range = X high − Xlow
- 20. RANGE CALCULATION • If the lowest observation in a sample of data were 20 and the highest was 75, then the range would be 55, or: • range = X high − X low 75 − 20 = 55.
- 21. RANGE AND ERROR CATCHING • Checking the ranges of variables is a good way to catch errors in a data set. • If a given measure, say a 7 point scale, is used and the range is found to be a 10 (e.g., a high score of 11 minus a low score of 1), some error occurred; logically, this range cannot exceed 6 (i.e., 7 – 1).
- 22. VARIANCE AND STANDARD DEVIATION • While the Mean is the chief measure of central tendency. • The standard deviation is the choice measure of dispersion.
- 23. VARIANCE • Variance is equal to the average of the squared deviations from the mean of a distribution. Symbolically, sample variance is 𝑠2 and the population variance 𝜎2 • Variance is the numerical index of variability, being based on the average of the squared deviations from the mean of a data set.
- 24. DEVIATION OF THE MEAN? ∑ (x - X‾) = 0 [ you will always get 0] [ when squared and summed, mean deviation scores yield positive numbers that are useful in a variety of statistical analyses] • The sum of squares (ss) is the sum of the squared deviations from the mean of a distribution. The ss is fundamental to descriptive and inferential statistics. SS = ∑ (X - X‾) 2
- 25. SUM OF SQUARES (SS) SS = ∑X 2 – (∑X) 2 /N Computational Formula
- 26. Greater variability means less consistency (larger deviations between X and X‾) in behavior, whereas less variability leads to greater consistency (smaller deviations between X and X‾)
- 27. SUM OF THE SQUARES • Influence • A) The size of the SS, then, is influenced by the magnitude of the deviation scores around the mean. • B) A second factor, the number of observations available, also plays a role in the SS’s size. • C) The more scores available in a distribution, the larger the SS will be, especially if some of the scores diverge greatly from the mean.
- 28. SAMPLE VARIANCE AND STANDARD DEVIATION 𝑠2 = ∑ (X - X‾) 2 /N 𝑠2 = SS/N Due to its derivation from ss, the 𝑠2 can be represented by Sample variance 𝑠2
- 29. STANDARD DEVIATION The standard deviation is the average deviation between an observed score and the mean of a distribution. Standard deviation, symbolized s, is determined by taking the square root of the variance, or √𝑠2 = s S = √𝑠2
- 30. STANDARD DEVIATION The standard deviation is the most common and, arguably, the best single index of a distribution’s variability. Low dispersion = homogeneous observations; high dispersion = heterogeneous observations.
- 31. 24 26 28 30 32 34 36 Y X A homogeneous Distribution A 30 2.0 B 72 5.0 Distrib. Mean SD 62 67 72 77 82 Y X A heterogeneous Distribution Figure 4.3 Comparing Distributions: Homogeneity and Heterogeneity
- 32. POPULATION VARIANCE AND STANDARD DEVIATION 𝜎2 = ∑ (X - 𝜇) 2 /N Or 𝜎2 = SS/N The population variance is the sum of the squared deviations between all observations (X) in the population and the mean of the population ( 𝜇) , which is divided by the total number of available observations.