module 13 introduction to normal probability distributions 6 of 19. here's what we know so far about normal probability curves the shape is a bell shape the mean is the center value associated with the peak the standard deviation measures the spread about the mean the standard deviation affects the shape of the normal curve the normal curve is always bell shaped but it can be short and squat or tall and skinny depending on the value of the standard deviation the area under the normal probability curve is equal to 1. normal probability curves also have another important feature the area under the curve is associated with values within one standard deviation of the mean is always 68 percent or 0.68 this is true for every normal curve regardless of the value of the mean or standard deviation we examine this important feature in the next example so example one standard deviation on each side of the mean let's start with a quantitative variable x that has a normal probability distribution with mean equal to 10 and standard deviation equal to 2. let's practice our new notation and mu is equal to 10 and sigma is equal to 2. so mu here right and the normal probability curve for x is shown below notice that since the standard deviation is equal to 2 so 10 plus 2 would give us 12 in other words the mean plus the standard deviation sigma or the mean minus one standard deviation ten minus two would give us eight as expected the mean mu equal to ten is located at the center of the normal curve right where you see the peak the other two arrows point to values a distance of one standard deviation on each side of the mean the point one standard deviation less than the mean is represented by mu minus sigma right since mu equals ten and sigma equals two this point is located at ten minus two which is equal to eight as shown the point 1 standard deviation more than the mean is represented by mu plus sigma since mu equals 10 and sigma equals 2 this point is located this point is located at 10 plus 2 which is equal to 12 as shown now you will notice we have indicated that the area of the green region is 68 percent or 0.68 so we can say that the probability of x being between 8 and 12 equals 0.68 or using probability notation we could write probability that x is between 8 and 10 or you know what we were saying above is equal to 68 probability of x is between 8 and 10. example baby birth weights baby birth weights have a normal probability distribution with mean mu equal to 3500 grams and standard deviation sigma equal to 600 grams what is the probability that a baby's birth weight is within one standard deviation of the mean birth weight well answer the probability is point 68 or 68 percent to be more precise there is a 68 chance that a randomly selected baby will weigh between 2 900 grams in other words 3 500 minus 600 equals 2900 and 4100 grams so mean of 3500 plus 600 for the standard deviation which is equal to 4100 at birth so let's summarize for any normal probability curve the central area within one standard deviation of the mean equals 0.68 this is the same as saying 68 percent of the time we will expect x to have a value within one standard deviation of the mean probability that x is between one standard deviation below and one standard deviation above is equal to 68 percent are you surprised by these statements after all we have seen that the standard deviation affects the shape of the normal curve the normal curve is always bell-shaped but sometimes it is short and squat or tall and skinny regardless the area under the normal probability curve is always one or one hundred percent same thing in addition the values within one standard deviation of the mean occur 68 of the time we examine this idea further in the next example so example inflection points look at the red arrows pointing at the normal curve in the following figure here are the red arrows at these points right here and over here right the curve changes the direction of its bend and goes from bending upward to bending downward or vice versa a point like this in a curve is called an inflection point every normal curve has inflection points at exactly one standard deviation on each side of the mean here's the mean standard deviation inflection point right standard deviation above and fraction point this diagram shows what we mean by concave down and concave up at and the transition between the two concave down so it's kind of like if it's the cup where would the water or coffee go down and concave up it's like part of a bowl here right the food is going to be above it so with the following applet you can look at a variety of normal curves so we're going to use the slider to change the standard deviation to get different normal curves observe that the two properties we discussed in the examples remain true for any standard deviation you select the probability that a value is within one standard deviation of the mean is 68 the x values of the inflection points correspond to one standard deviation above and below the mean okay so here's the applet and notice it says that the mean is 10 and standard deviation is right now it's at 4. so we have 10 plus 4 14 10 minus 4 6. but what happens to the shape if we move the standard deviation look how we can make it really really really skinny right and move the standard deviation to be 1.12 there let's see move it all the way to 1 and if you notice right normal curve mean is 10 same same mean but the standard deviation changes and notice how the shape changed but the area in the like purple pink inside is still 68 or 0.68 right so it doesn't matter how spread out our curve is that part of one standard deviation above and below always stays at 68 percent