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HELPFUL VOCABULARY Normal distribution: a symmetrical bellshaped curve with tails that extend infinitely in both directions from the mean of a data set. Standard deviation: A measure of spread of a data set equal to the square root of the sum of the squared variances divided by the number of data values. NORMAL DISTRIBUTION NORMAL DISTRIBUTION In a normal curve, the right side of the curve is a mirror image of the left side. The mean of a normal distribution is the center or balance point of the two sides (point where the curve is the highest). STANDARD DEVIATION The data spreads from the center or mean of the curve. This spread is called deviation. STANDARD DEVIATION STANDARD DEVIATION HELPFUL VOCABULARY Inflection point: A point on a normal curve where it goes from being concave down to being concave up. On a normal curve, inflection points occur at 1 standard deviation from the mean. 68-95-99.7 Rule: The rule that includes the percentages of data that are within 1, 2, and 3 standard deviations of the mean of a set of data. Variance: A measure of spread of a data set equal to the mean of the squared variations of each data value from the mean of the data set. STANDARD DEVIATION Inflection points: where the graph goes from concave down to concave up. The points are the same distance from the mean on both sides of the curve. The inflection points are one standard deviation away from the mean. Approximately 68% of all data will fall within one standard deviation of the mean. 68-95-99.7 RULE 68% of all data values should be located within 1 standard deviation of the mean 95% of all data values should be located within 2 standard deviations of the mean 99.7% of all data values should be located (X-µ) within 3 standard deviations of the mean CALCULATING STANDARD DEVIATION Calculate the standard deviation of the following numbers, which represent a small population. 2, 7, 5, 6, 4, 2, 6, 3, 6, 9 1. Find the mean of the numbers. 2. Fill out the chart below. X (X - µ) (X - µ)² CALCULATING STANDARD DEVIATION Mean= 5.0 Standard deviation Formula: ( − )2 = X (X - µ) (X - µ)² 2 2 – 5 = -3 (-3)² = 9 7 7–5=2 (2)² = 4 5 5–5=0 (0)² = 0 6 6–5=1 (1)² = 1 4 4 – 5 = -1 (-1)² = 1 2 2 – 5 = -3 (-3)² = 9 6 6–5=1 (1)² = 1 3 3 – 5 = -2 (-2)² = 4 6 6–5=1 (1)² = 1 9 9–5=4 (4)² = 16 (9 + 4 + 0 + 1 + 1 + 9 + 1 + 4 + 1 + 16) 10 CALCULATING STANDARD DEVIATION Find the standard deviation of the following two sets of numbers. 40, 50, 35, 40, 45, 30 15, 65, 55, 35, 45, 25 What do you notice about the means of the data? What do you notice about the standard variation of the data? VARIANCE Variance of a set of data is the square of the standard deviation. Variance = σ² Standard Deviation = σ NORMAL DISTRIBUTION CURVE EXAMPLE The lifetimes of a certain type of light bulb are normally distributed. The mean life is 400 hours, and the standard deviation is 75 hours. For a group of 5,000 light bulbs, how many are expected to last each of the following times? Between 325 hours and 475 hours More than 250 hours Less than 250 hours EXAMPLE A bag of chips has a mean mass of 70 g, with a standard deviation of 3 g. Assuming a normal distribution, create a normal curve including all necessary values. If 1,250 bags of chips are processed each day, how many bags will have a mass between 67 and 73 g? What percentage of the bags of chips will have a mass greater than 64? Z-SCORE A measure of the number of standard deviations a particular data point is away from the mean of the sample data. − = EXAMPLE The grades on a statistics mid-term for a high school are normally distributed, with a mean of 81, standard deviation of 6.3. Calculate the z-scores for each of the following exam grades: 65, 83, 93, 100. -2.54 .32 1.90 3.02 EXAMPLE A selective college only admits students who place at least 2.5 z-scores above the mean on the ACT with a mean of 19 and a standard deviation of 5. What is the minimum score that an applicant must obtain to be admitted to the university? PRACTICE Z-Score Practice Worksheet - Work on this for the rest of the class period; due tomorrow. PERCENTAGES A z- score chart tells you the percentage of area BELOW (to the left) of your given z-score. Need to know the mean and standard deviation. EXAMPLE If z = 1, then 84.3% of the data is below that given value. If z = 1, then 15.87% of the data is above that given value. CHART Look at the chart given to you. Find a z-score of -1.45. What percentage of area is BELOW this z-score? .0735 or 7.35% CHART Look at the chart given to you. Find a z-score of 3.19. What percentage of area is BELOW this z-score? What percentage of area is ABOVE this z-score? .9993 or 99.93% .0007 or 0.07% EXAMPLE On a nationwide math test, the mean was 65 and the standard deviation was 10. If Robert scored 81, what was his z-score? What percentage of students scored below Robert? What percentage of students scored above Robert? FINDING AREA TO THE RIGHT AND LEFT OF A ZSCORE LEFT: Find the z-score on the chart; that is your area. RIGHT Find the z-score on the chart; subtract from 1. FINDING AREA BETWEEN TWO Z-SCORES Find the area of the whole and subtract the area of the smaller portion. Z-SCORES AND PROBABILITY Find the probability: P(0 < z < 1.25) P( -1.3 < z < 2) .3944 or 39.44% .8804 or 88.04%