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MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets. MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets. The histogram shown gives an idea of the shape of a normal distribution. MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets. Although the normal distribution is a continuous distribution whose graph is a smooth curve, an appropriate histogram can give a very good approximation to the actual normal graph. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. 4. The curve is symmetric with respect to its mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. 4. The curve is symmetric with respect to its mean. 5. The total area under the curve is 1. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. 4. The curve is symmetric with respect to its mean. 5. The total area under the curve is 1. 6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. 4. The curve is symmetric with respect to its mean. 5. The total area under the curve is 1. 6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. 4. The curve is symmetric with respect to its mean. 5. The total area under the curve is 1. 6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal Distribution π is the mean and π is the standard deviation of the distribution. 1. A normal curve is bell-shaped. 2. The highest point on the curve is at the mean. 3. The mean, median and mode are equal. 4. The curve is symmetric with respect to its mean. 5. The total area under the curve is 1. 6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean. The 68-95-99.7 Rule for Normal Distributions MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. 425 475 MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. 425 475 MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. 425 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) 425 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) 425 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) 450 β 425 = 25 25 425 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) 475 β 450 = 25 25 25 425 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) 25 425 25 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) But 25 is the standard deviation. 25 25 425 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean. Here π = πππ(mean) and π = ππ(std deviation) The interval shown consists of all scores within 1 standard deviation of the mean. But 25 is the standard deviation. 25 25 425 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95%68-95-99.7 are within 2Rule standard of the mean and The tellsdeviations us that 68% of the scores 99.7% are with 3 standard deviations themean. mean. are within 1 standard deviation ofofthe Here π = πππ(mean) and π = ππ(std deviation) The interval shown consists of all scores within 1 standard deviation of the mean. 25 425 25 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95%68-95-99.7 are within 2Rule standard of the mean and The tellsdeviations us that 68% of the scores 99.7% are with 3 standard deviations themean. mean. are within 1 standard deviation ofofthe Here π = πππ(mean) and π = ππ(std deviation) The interval shown consists of all scores within 1 standard deviation of the mean. 68% 25 425 25 450 475 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95%68-95-99.7 are within 2Rule standard of the mean and The tellsdeviations us that 68% of the scores 99.7% are with 3 standard deviations themean. mean. are within 1 standard deviation ofofthe Here π = πππ(mean) and π = ππ(std deviation) The interval shown consists of all scores within 1 standard deviation of the mean. 68% 25 25 If there are 1000 scores, we would expect about 425 475 68% of them to be between 425 and 475. 450 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95%68-95-99.7 are within 2Rule standard of the mean and The tellsdeviations us that 68% of the scores 99.7% are with 3 standard deviations themean. mean. are within 1 standard deviation ofofthe Here π = πππ(mean) and π = ππ(std deviation) The interval shown consists of all scores within 1 standard deviation of the mean. 68% 25 25 If there are 1000 scores, we would expect about 425 475 68% of them to be between 425 and 475. 450 1000 68% = 1000 0.68 = 680 The shaded area gives the probability of a score falling in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall between 425 and 475? The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean, 95%68-95-99.7 are within 2Rule standard of the mean and The tellsdeviations us that 68% of the scores 99.7% are with 3 standard deviations themean. mean. are within 1 standard deviation ofofthe Here π = πππ(mean) and π = ππ(std deviation) The interval shown consists of all scores within 1 standard deviation of the mean. 68% 25 25 If there are 1000 scores, we would expect about 425 475 68% of them to be between 425 and 475. 450 1000 68% = 1000 0.68 = 680 So, we expect about 680 scores to be in the 425 β 475 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. 450 MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. Only the question has changed. 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 500 is from the mean. 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 500 is from the mean. 500 β 450 = 50 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 500 is from the mean. But 25 is the standard deviation. 500 β 450 = 50 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 500 is from the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. But 25 is the standard deviation. 500 β 450 = 50 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This is exactly the same distribution as before. The 68-95-99.7 Rule 68% of the values within 1has std. deviation Only theare question changed.of the mean, 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule within 2 standard of theofmean. 68%are of the values are withindeviations 1has std. deviation the mean, Only the question changed. 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule within 2 standard of theofmean. 68%are of the values are withindeviations 1has std. deviation the mean, Only the question changed. 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 95% 50 400 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule That leaves are within 2 standard deviations of the mean. 68% of the values within 1has std. deviation Only theare question changed.of the mean, 5% to be split 95%we areneed withinto 2 standard of the mean and between the First find howdeviations many standard two tails. 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 95% 50 400 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule That leaves are within 2 standard deviations of the mean. 68% of the values within 1has std. deviation Only theare question changed.of the mean, 5% to be split 95%we areneed withinto 2 standard of the mean and between the First find howdeviations many standard two tails. 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. Half of 5% is 2.5%. So the orange shaded area is 2.5%. 95% 50 400 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule That leaves are within 2 standard deviations of the mean. 68% of the values within 1has std. deviation Only theare question changed.of the mean, 5% to be split 95%we areneed withinto 2 standard of the mean and between the First find howdeviations many standard two tails. 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 95% 2.5% Half of 5% is 2.5%. So the orange shaded area is 2.5%. 50 400 2.5% 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule That leaves are within 2 standard deviations of the mean. 68% of the values within 1has std. deviation Only theare question changed.of the mean, 5% to be split 95%we areneed withinto 2 standard of the mean and between the First find howdeviations many standard two tails. 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 95% 2.5% Half of 5% is 2.5%. So the orange shaded area is 2.5%. 1000 2.5% = 1000 0.025 =25 50 400 2.5% 50 450 500 The shaded area gives the probability of a score falling above 500. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall above 500? This exactly the same asthe before. Theis68-95-99.7 Rule tellsdistribution us that 95% of scores The 68-95-99.7 Rule That leaves are within 2 standard deviations of the mean. 68% of the values within 1has std. deviation Only theare question changed.of the mean, 5% to be split 95%we areneed withinto 2 standard of the mean and between the First find howdeviations many standard two tails. 99.7% are500 withis3from standard deviations thedeviations mean. of the mean. So 500 is 2 standard deviations (25 + 25 = 50) from the mean. 95% 2.5% Half of 5% is 2.5%. So the orange shaded area is 2.5%. 1000 2.5% = 1000 0.025 =25 50 400 2.5% 50 450 500 So,shaded we expect 25 scores to beof The areaabout gives the probability a score fallingabove in the500. 450 β 500 range. MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 1 standard deviation from the mean 68% of scores πβπ π π+π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 1 standard deviation from the mean 68% of scores 34% 34% πβπ π π+π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 1 standard deviation from the mean 16% 68% of scores 34% 34% πβπ π 16% π+π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 2 standard deviations from the mean 95% of scores π β 2π π π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 2 standard deviations from the mean 95% of scores 47.5% 47.5% π β 2π π π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 2 standard deviations from the mean 2.5% 95% of scores 47.5% 47.5% π β 2π π 2.5% π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 3 standard deviations from the mean 99.7% of scores π β 3π π π + 3π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 3 standard deviations from the mean 99.7% of scores 49.85% 49.85% π β 3π π π + 3π MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve. When endpoints are 3 standard deviations from the mean 0.15% π β 3π 99.7% of scores 49.85% 49.85% π 0.15% π + 3π When endpoints are 1 standard deviation from the mean SUMMARY How each percentage of the 68-95-99.7 rule breaks down underneath the Normal Curve 16% 68% of scores 34% 34% 16% πβπ π π+π When endpoints are 2 standard deviations from the mean 2.5% 95% of scores 47.5% 47.5% 2.5% π β 2π π π + 2π When endpoints are 3 standard deviations from the mean 0.15% π β 3π 99.7% of scores 49.85% 49.85% π 0.15% π + 3π MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. 450 MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. Only the question has changed. 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 375 is from the mean. 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 375 is from the mean. 450 β 375 = 75 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 375 is from the mean. But 25 is the standard deviation. 450 β 375 = 75 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. Only the question has changed. First we need to find how many standard deviations 375 is from the mean. So 375 is 3 standard deviations (25 + 25 + 25 =75) from the mean. But 25 is the standard deviation. 450 β 375 = 75 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. The 68-95-99.7 Rule 68% of the values within 1has std. deviation Only theare question changed.of the mean, 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard 99.7% are375 withis3from standard deviations thedeviations mean. of the mean. So 375 is 3 standard deviations (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This is exactly the same distribution as before. The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. are within 3 standard deviations of the mean. 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard 99.7% are375 withis3from standard deviations thedeviations mean. of the mean. So 375 is 3 standard deviations (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. The 68-95-99.7 Rule letβs take The 68-95-99.7 Rule tells us that 99.7% of the scores 68% of the values within 1has std. deviation Only theare question changed.of the mean, advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations thedeviations mean. of the mean. developed. So 375 is 3 standard deviations (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. The 68-95-99.7 Rule letβs take The 68-95-99.7 Rule tells us that 99.7% of the scores 68% of the values within 1has std. deviation Only theare question changed.of the mean, advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean and First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations thedeviations mean. of the mean. developed. So 375 is 3 standard deviations (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. When endpoints are 1 standard deviation from the mean SUMMARY How each percentage of the 68-95-99.7 rule breaks down underneath the Normal Curve 16% 68% of scores 34% 34% 16% πβπ π π+π When endpoints are 2 standard deviations from the mean 2.5% 95% of scores 47.5% 47.5% 2.5% π β 2π π π + 2π When endpoints are 3 standard deviations from the mean 0.15% π β 3π 99.7% of scores 49.85% 49.85% π 0.15% π + 3π When endpoints are 1 standard deviation from the mean SUMMARY How each percentage of the 68-95-99.7 rule breaks down underneath the Normal Curve 16% 68% of scores 34% 34% 16% πβπ π π+π When endpoints are 2 standard We figured out that 375 is deviations from the mean 3 standard deviations of scores from 95% the mean, so the graph is 47.5% the one 2.5% 2.5% bottom 47.5% π β 2π that weπneed. π + 2π When endpoints are 3 standard deviations from the mean 0.15% π β 3π 99.7% of scores 49.85% 49.85% π 0.15% π + 3π MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. When endpoints are 3 standardletβs take The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. deviations from the mean advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean andof scores 99.7% First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations mean. 49.85% 49.85% deviations the0.15% mean. of the 0.15% developed. So 375 is 3 standard deviations π β 3π π π + 3π (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. When endpoints are 3 standardletβs take The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. deviations from the mean advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean andof scores 99.7% First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations mean. 49.85% 49.85% deviations the0.15% mean. of the 0.15% developed. So 375 is 3 standard deviations π β 3π π π + 3π (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. When endpoints are 3 standardletβs take The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. deviations from the mean advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean andof scores 99.7% First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations mean. 49.85% 49.85% deviations the0.15% mean. of the 0.15% developed. So 375 is 3 standard deviations π β 3π π π + 3π (25 + 25 + 25 =75) from the mean. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. When endpoints are 3 standardletβs take The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. deviations from the mean advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean andof scores 99.7% First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations mean. 49.85% 49.85% deviations the0.15% mean. of the 0.15% developed. So 375 is 3 standard deviations π β 3π π π + 3π (25 + 25 + 25 =75) from the mean. So the orange shaded area is 0.15%. 75 375 450 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. When endpoints are 3 standardletβs take The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. deviations from the mean advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean andof scores 99.7% First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations mean. 49.85% 49.85% deviations the0.15% mean. of the 0.15% developed. So 375 is 3 standard deviations π β 3π π π + 3π (25 + 25 + 25 =75) from the mean. So the orange shaded area is 0.15%. 75 450 1000 0.15% = 1000 0.0015 =1.5 375 The shaded area gives the probability of a score falling below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. How many scores do we expect to fall below 375? This time, This is exactly the same distribution as before. When endpoints are 3 standardletβs take The 68-95-99.7 Rule The tells us1has thatdeviation 99.7% ofofthe 68% 68-95-99.7 of the values are within std. thescores mean, Only theRule question changed. deviations from the mean advantage of are within 3 standard deviations of the mean. the summary 95%we areneed withinto 2 standard of the mean andof scores 99.7% First find howdeviations many standard sheet we 99.7% are375 withis3from standard deviations mean. 49.85% 49.85% deviations the0.15% mean. of the 0.15% developed. So 375 is 3 standard deviations π β 3π π π + 3π (25 + 25 + 25 =75) from the mean. So the orange shaded area is 0.15%. 1000 0.15% = 1000 0.0015 =1.5 375 75 450 So, we expect about 1.5 scores to be below 375. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. First, we must decide which part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. To decide the percent to use, we must find the number of standard deviations there are between 180 and the mean (150). First, we must decide which part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. To decide the percent to use, we must find the number of standard deviations there are between 180 and the mean (150). 180 β 150 = 30 180 β 150 = 30 180 β 150 = 30 First, we must decide which part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. To decide the percent to use, we must find the number of standard deviations there are between 180 and the mean (150). 180 β 150 = 30 30 (10+10+10) is 3 standard deviations First, we must decide which part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. To decide the percent to use, we must find the number of standard deviations there are between 180 and the mean (150). 180 β 150 = 30 So, 180 is 3 standard deviations from the mean. 30 (10+10+10) is 3 standard deviations First, we must decide which part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? Given what we have learned, letβs try to solve this as efficiently as possible. To decide the percent to use, we must find the number of standard deviations there are between 180 and the mean (150). 180 β 150 = 30 So, 180 is 3 standard deviations from the mean. First, we must decide which part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? To decide the percent to use, we Given what endpoints 3 standard find theare number of standard we haveWhenmust deviations are between fromthere the mean learned, letβs deviations 180 and mean (150). try to solve 99.7% ofthe scores this as 49.85% 49.85% 0.15% 180 β 150 = 30 0.15% efficiently as π β 3π So, 180 isπ3 standard deviations π + 3π possible. from the mean. So, the 3 standard deviation part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? To decide the percent to use, we Given what endpoints 3 standard find theare number of standard we haveWhenmust deviations are between fromthere the mean learned, letβs deviations 180 and mean (150). try to solve 99.7% ofthe scores this as 49.85% 49.85% 0.15% 180 β 150 = 30 0.15% efficiently as π β 3π So, 180 isπ3 standard deviations π + 3π possible. from 150 the mean. 180 So, the 3 standard deviation part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? To decide the percent to use, we Given what endpoints 3 standard find theare number of standard we haveWhenmust deviations are between fromthere the mean learned, letβs deviations 180 and mean (150). try to solve 99.7% ofthe scores this as 49.85% 49.85% 0.15% 180 β 150 = 30 0.15% efficiently as π β 3π So, 180 isπ3 standard deviations π + 3π possible. from 150 the mean. 180 49.85% is the percent between 150 and 180. So, the 3 standard deviation part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? To decide the percent to use, we Given what endpoints 3 standard find theare number of standard we haveWhenmust deviations are between fromthere the mean learned, letβs deviations 180 and mean (150). try to solve 99.7% ofthe scores this as 49.85% 49.85% 0.15% 180 β 150 = 30 0.15% efficiently as π β 3π So, 180 isπ3 standard deviations π + 3π possible. from 150 the mean. 180 49.85% is the percent between 150 and 180. 2000 49.85% = 2000 0.4985 = 997 So, the 3 standard deviation part of the β68-95-99.7 Ruleβ applies. MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 2000 students on an intelligence test is a normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180? To decide the percent to use, we Given what endpoints 3 standard find theare number of standard we haveWhenmust deviations are between fromthere the mean learned, letβs deviations 180 and mean (150). try to solve 99.7% ofthe scores this as 49.85% 49.85% 0.15% 180 β 150 = 30 0.15% efficiently as π β 3π So, 180 isπ3 standard deviations π + 3π possible. from 150 the mean. 180 49.85% is the percent between 150 and 180. 2000 49.85% = 2000 0.4985 = 997 So, the 3 standard deviation part of the β68-95-99.7 Ruleβ applies. So, we expect about 997 scores to be between 150 and 180. MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. Step 3. Use the percent diagram for the β2 standard deviationβ case. MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. Step 3. Use the percent diagram for the β2 standard deviationβ case. When endpoints are 2 standard deviations from the mean 2.5% 95% of scores 47.5% 47.5% π β 2π π 2.5% π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. Step 3. Use the percent diagram for the β2 standard deviationβ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that 95% of scores goes with βbelow 26β. 2.5% 47.5% π β 2π π 47.5% 2.5% π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. Step 3. Use the percent diagram for the β2 standard deviationβ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that 95% of scores goes with βbelow 26β. 26 2.5% 47.5% π β 2π π 47.5% 2.5% π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. Step 3. Use the percent diagram for the β2 standard deviationβ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that 95% of scores goes with βbelow 26β. 26 2.5% 47.5% π β 2π π 47.5% 2.5% π + 2π MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5. What percentage of values do we expect to be below 26? Letβs use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). ππ β ππ = ππ Step 2. Express that difference in terms of standard deviations. Standard deviation is 5 so 10 is 2 standard deviations from the mean. Step 3. Use the percent diagram for the β2 standard deviationβ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that 95% of scores goes with βbelow 26β. 26 2.5% 47.5% 47.5% 2.5% π β 2π π π + 2π Step 5. Answer: About 2.5% of values will be below 26.