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 1. A normal curve is bell-shaped.

MATH 110 Sec 14-4 Lecture: The Normal Distribution Properties of the Normal 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 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 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 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 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 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. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 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. 425475 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 The shaded area gives the probability of a score falling in the 425 – 475 range. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 450 425475 The shaded area gives the probability of a score falling in the 425 – 475 range. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 450 425475 The shaded area gives the probability of a score falling in the 425 – 475 range. 450 – 425 = 25 25 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 450 425475 The shaded area gives the probability of a score falling in the 425 – 475 range. 475 – 450 = 25 25 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 450 425475 The shaded area gives the probability of a score falling in the 425 – 475 range. 25 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 450 425475 The shaded area gives the probability of a score falling in the 425 – 475 range. 25 But 25 is the standard deviation. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 But 25 is the standard deviation. The interval shown consists of all scores within 1 standard deviation of the mean. 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. The interval shown consists of all scores within 1 standard deviation of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. 68% The interval shown consists of all scores within 1 standard deviation of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. 68% The interval shown consists of all scores within 1 standard deviation of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475? About 68% of scores should fall between 425 & 475.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. 68% The interval shown consists of all scores within 1 standard deviation of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475? About 68% of scores should fall between 425 & 475. How many of the 1000 scores would we expect to be between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. 68% The interval shown consists of all scores within 1 standard deviation of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475? If there are 1000 scores, we would expect about 68% of them to be between 425 and 475. About 68% of scores should fall between 425 & 475. How many of the 1000 scores would we expect to be between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 The shaded area gives the probability of a score falling in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. 68% The interval shown consists of all scores within 1 standard deviation of the mean. If there are 1000 scores, we would expect about 68% of them to be between 425 and 475. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475? About 68% of scores should fall between 425 & 475. How many of the 1000 scores would we expect to be between 425 & 475?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 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. 425475 25 450 So, we expect about 680 scores to be in the 425 – 475 range. The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean. 68% The interval shown consists of all scores within 1 standard deviation of the mean. If there are 1000 scores, we would expect about 68% of them to be between 425 and 475. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475? About 68% of scores should fall between 425 & 475. How many of the 1000 scores would we expect to be between 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. What percent of scores would be expected to fall above 500?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500?

The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 500 The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 500 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 500 – 450 = 50 50 500 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 500 – 450 = 50 50 But 25 is the standard deviation. 500 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 500 – 450 = 50 50 But 25 is the standard deviation. 500 The shaded area is the probability of a score being above 500.

The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. Half of 5% is 2.5%. So the orange shaded area is 2.5%. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? The shaded area is the probability of a score being above 500.

The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. 2.5% Half of 5% is 2.5%. So the orange shaded area is 2.5%. The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. 2.5% The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? About 2.5% of scores should be above 500. Half of 5% is 2.5%. So the orange shaded area is 2.5%. The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. 2.5% The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? About 2.5% of scores should be above 500. How many of the 1000 scores would we expect to be above 500? Half of 5% is 2.5%. So the orange shaded area is 2.5%. The shaded area is the probability of a score being above 500.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. 2.5% The shaded area is the probability of a score being above 500. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? About 2.5% of scores should be above 500. We expect about 2.5% of them to be above 500. Half of 5% is 2.5%. So the orange shaded area is 2.5%. How many of the 1000 scores would we expect to be above 500?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. First we need to find how many standard deviations 500 is from the mean. Only the question has changed. 50 500 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. The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean. 95% 50 400 That leaves 5% to be split between the two tails. 2.5% The shaded area is the probability of a score being above 500. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500? About 2.5% of scores should be above 500. We expect about 2.5% of them to be above 500. Half of 5% is 2.5%. So the orange shaded area is 2.5%. How many of the 1000 scores would we expect to be above 500? So, we expect about 25 scores to be above 500.

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

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

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

When endpoints are 2 standard deviations from the mean 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

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. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. The shaded area gives the probability of a score falling below 375. 375 The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 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 The shaded area gives the probability of a score falling below 375. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. 450 – 375 = 75 75 The shaded area gives the probability of a score falling below 375. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. 450 – 375 = 75 The shaded area gives the probability of a score falling below 375. 75 But 25 is the standard deviation. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. The shaded area gives the probability of a score falling below 375. 75 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. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. The shaded area gives the probability of a score falling below 375. 75 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. The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. The shaded area gives the probability of a score falling below 375. 75 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. The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean. This time, let’s take advantage of the summary sheet we developed. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

When endpoints are 3 standard deviations from the mean When endpoints are 2 standard deviations from the mean 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

When endpoints are 3 standard deviations from the mean When endpoints are 2 standard deviations from the mean 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 We figured out that 375 is 3 standard deviations from the mean, so the bottom graph is the one that we need.

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. The shaded area gives the probability of a score falling below 375. 75 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. The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean. This time, let’s take advantage of the summary sheet we developed. When endpoints are 3 standard deviations from the mean The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. The shaded area gives the probability of a score falling below 375. 75 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. The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean. This time, let’s take advantage of the summary sheet we developed. When endpoints are 3 standard deviations from the mean So the orange shaded area is 0.15%. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

MATH 110 Sec 14-4 Lecture: The Normal Distribution 450 This is exactly the same distribution as before. Only the question has changed. 375 First we need to find how many standard deviations 375 is from the mean. The shaded area gives the probability of a score falling below 375. 75 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. The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean. This time, let’s take advantage of the summary sheet we developed. When endpoints are 3 standard deviations from the mean So the orange shaded area is 0.15%. The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375? So, we expect about 0.15% of the 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, let’s 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, let’s try to solve this as efficiently as possible. First, we must decide which part of the ‘68-95-99.7 Rule’ 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, let’s 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 ‘68-95-99.7 Rule’ 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, let’s 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). 30 (10+10+10) is 3 standard deviations First, we must decide which part of the ‘68-95-99.7 Rule’ 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, let’s 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). 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 ‘68-95-99.7 Rule’ 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, let’s 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). So, 180 is 3 standard deviations from the mean. First, we must decide which part of the ‘68-95-99.7 Rule’ 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, let’s 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). So, 180 is 3 standard deviations from the mean. So, the 3 standard deviation part of the ‘68-95-99.7 Rule’ applies. When endpoints are 3 standard deviations from the mean

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, let’s 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). So, 180 is 3 standard deviations from the mean. So, the 3 standard deviation part of the ‘68-95-99.7 Rule’ applies. When endpoints are 3 standard deviations from the mean 150180

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, let’s 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). So, 180 is 3 standard deviations from the mean. So, the 3 standard deviation part of the ‘68-95-99.7 Rule’ applies. When endpoints are 3 standard deviations from the mean 150180 49.85% is the percent 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, let’s 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). So, 180 is 3 standard deviations from the mean. So, the 3 standard deviation part of the ‘68-95-99.7 Rule’ applies. When endpoints are 3 standard deviations from the mean 150180 49.85% is the percent between 150 and 180. 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? Let’s 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? Let’s 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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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 ‘2 standard deviation’ 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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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 ‘2 standard deviation’ case. When endpoints are 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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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 ‘2 standard deviation’ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that goes with ‘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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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 ‘2 standard deviation’ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that goes with ‘below 26’. 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? Let’s use this exercise to try to get the answer while showing even less work. Step 1. Find the difference between 26 and 36 (mean). 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 ‘2 standard deviation’ case. When endpoints are 2 standard deviations from the mean Step 4. Find the percent that goes with ‘below 26’. 26 About 2.5% of values will be below 26. Step 5. Answer:

MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets.

Similar presentations

Presentation on theme: "MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets.

Similar presentations

Presentation on theme: "MATH 110 Sec 14-4 Lecture: The Normal Distribution The normal distribution describes many real-life data sets."— Presentation transcript:

Similar presentations

About project

Feedback