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Chapter 5 NORMAL DISTRIBUTION AND Z-SCORE Understanding Statistics for International Social Work and Other Behavioral Sciences Serge Lee, Maria C Silveira Nunes Dinis, Lois Lowe, and Kelly Anders (2015). Oxford University Press NORMAL CURVE 2 The sum of the deviations of all observation scores from the mean is balanced at the center point on the number line, which is equal to or very near zero SKEWED DISTRIBUTION 3 Positively Skewed Distributions Negatively Skewed Distributions Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press STANDARD DEVIATION ON PROPORTION OF THE NORMAL CURVE 4 The normal curve is divided into six equal units. The six units collectively reflect the variation that exists within virtually all values of a normally distributed interval or ratio level scale variable. Confidence interval Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press DISTANCE BETWEEN THE SAMPLE MEAN AND STANDARD DEVIATION 5 The proportion between the mean and their corresponding standard deviations are divided as below: Distance from the mean and standard deviation Left side only of the curve between mean and SD (-SD) Right side only of the curve between mean and SD (+SD) Both sides under areas of the normal curve 1SD 34.13 34.13 68.26 2SD 13.59 13.59 27.18 3SD 2.15 2.15 4.30 Total areas under the normal curve 49.87 49.87 99.74 Confidence interval Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press CALCULATE THE Z-SCORE AND CONVERT TO A PERCENTILE 6 Formula: Z= X−X SD Statistics rules for the conversion of the Z-score into percentile ranks: • Always round the calculated Z-score to two decimal places. For example, Z = -2.29 • If the calculated Z-score is positive (+), add 50% or .50 (if use decimals) to the percentile obtained from the standard normal curve table • If the calculated Z-score is negative (-), subtract the percentile obtained from the critical values of the standard normal curve table from 50% or .50 (if use decimals)