Econ 160: Introduction to Statistics Myles J Callan Spring 2009
The following are the list of some of its important properties of the normal distribution.
1. The mean, median and mode have the same value and this value is located exactly at the center of the distribution.
2. The normal curve is symmetrical about the mean.
3. The tails of the curve do not touch the horizontal axis, and will not touch it no matter how far away they may be
extended.
4. The total area under the normal curve equals 1. The area to the left of the mean is .5 (or 50%) and the area to the
right of the mean is also .5 (50%). That is, we expect 50% of the data to be below the mean, and 50% above the
mean.
5. Data that lie beyond three standard deviations from the mean are very rare, although, the range of a true normal
distribution is infinite.
The area under the curve within a certain interval on the horizontal axis is equal to the probability that a value selected
at random, from a set of numbers that has this distribution, will lie in that interval.
Additional Properties of the normal curve: The Empirical rules
Given any normal distribution, we can obtain a more exact statement concerning the proportion of t he items within
specific standard deviations from the mean.
The empirical rule is given as follows:
1. About 68% of the items will lie within the interval from one standard deviation below the mean to one standard
deviation above the mean, that is, within the interval )(
σµσµ
+− to .
2. About 95% of the items will lie within the interval from two standard deviations below the mean to two standard
deviations above the mean, that is, within the interval
)22(
σµσµ
+− to
.
3. Almost all (99.7%) of the items will lie within the interval from three standard deviations below the mean to three
standard deviations above the mean, that is, within the interval
)33(
σµσµ
+− to
.
The empirical rule can be illustrated on a normal distribution.
Z scores and the standard normal curve
The z score (or standard units) is a measure of the relative position of a data item in terms of the number of standard
deviations it is from the mean. We can transform each original value into standard units, z, by using the formula,
z =
deviation
standard
mean valueoriginal
−
z =
s
xx
−
for sample data or z =
σ
µ
−
x
for population data.
z
zz
z
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0
00
0
0 0.008 0.016 0.024 0.032 0.0398 0.0478 0.0558 0.0638 0.0718
0.1
0.10.1
0.1
0.0796 0.0876 0.0956 0.1034 0.1114 0.1192 0.1272 0.135 0.1428 0.1506
0.2
0.20.2
0.2
0.1586 0.1664 0.1742 0.182 0.1896 0.1974 0.2052 0.2128 0.2206 0.2282
0.3
0.30.3
0.3
0.2358 0.2434 0.251 0.2586 0.2662 0.2736 0.2812 0.2886 0.296 0.3034
0.4
0.40.4
0.4
0.3108 0.3182 0.3256 0.3328 0.34 0.3472 0.3544 0.3616 0.3688 0.3758
0.5
0.50.5
0.5
0.383 0.39 0.397 0.4038 0.4108 0.4176 0.4246 0.4314 0.438 0.4448
0.6
0.60.6
0.6
0.4514 0.4582 0.4648 0.4714 0.4778 0.4844 0.4908 0.4972 0.5034 0.5098
0.7
0.70.7
0.7
0.516 0.5222 0.5284 0.5346 0.5408 0.5468 0.5528 0.5588 0.5646 0.5704
0.8
0.80.8
0.8
0.5762 0.582 0.5878 0.5934 0.599 0.6046 0.6102 0.6156 0.6212 0.6266
0.9
0.90.9
0.9
0.6318 0.6372 0.6424 0.6476 0.6528 0.6578 0.663 0.668 0.673 0.6778
1
11
1
0.6826 0.6876 0.6922 0.697 0.7016 0.7062 0.7108 0.7154 0.7198 0.7242
1.1
1.11.1
1.1
0.7286 0.733 0.7372 0.7416 0.7458 0.7498 0.754 0.758 0.762 0.766
1.2
1.21.2
1.2
0.7698 0.7738 0.7776 0.7814 0.785 0.7888 0.7924 0.796 0.7994 0.803
1.3
1.31.3
1.3
0.8064 0.8098 0.8132 0.8164 0.8198 0.823 0.8262 0.8294 0.8324 0.8354
1.4
1.41.4
1.4
0.8384 0.8414 0.8444 0.8472 0.8502 0.853 0.8558 0.8584 0.8612 0.8638
1.5
1.51.5
1.5
0.8664 0.869 0.8714 0.874 0.8764 0.8788 0.8812 0.8836 0.8858 0.8882
1.6
1.61.6
1.6
0.8904 0.8926 0.8948 0.8968 0.899 0.901 0.903 0.905 0.907 0.909
1.7
1.71.7
1.7
0.9108 0.9128 0.9146 0.9164 0.9182 0.9198 0.9216 0.9232 0.925 0.9266
1.8
1.81.8
1.8
0.9282 0.9298 0.9312 0.9328 0.9342 0.9356 0.9372 0.9386 0.9398 0.9412
1.9
1.91.9
1.9
0.9426 0.9438 0.9452 0.9464 0.9476 0.9488 0.95 0.9512 0.9522 0.9534
2
22
2
0.9544 0.9556 0.9566 0.9576 0.9586 0.9596 0.9606 0.9616 0.9624 0.9634
2.1
2.12.1
2.1
0.9642 0.9652 0.966 0.9668 0.9676 0.9684 0.9692 0.97 0.9708 0.9714
2.2
2.22.2
2.2
0.9722 0.9728 0.9736 0.9742 0.975 0.9756 0.9762 0.9768 0.9774 0.978
2.3
2.32.3
2.3
0.9786 0.9792 0.9796 0.9802 0.9808 0.9812 0.9818 0.9822 0.9826 0.9832
2.4
2.42.4
2.4
0.9836 0.984 0.9844 0.985 0.9854 0.9858 0.9862 0.9864 0.9868 0.9872
2.5
2.52.5
2.5
0.9876 0.988 0.9882 0.9886 0.989 0.9892 0.9896 0.9898 0.9902 0.9904
2.6
2.62.6
2.6
0.9906 0.991 0.9912 0.9914 0.9918 0.992 0.9922 0.9924 0.9926 0.9928
2.7
2.72.7
2.7
0.993 0.9932 0.9934 0.9936 0.9938 0.994 0.9942 0.9944 0.9946 0.9948
2.8
2.82.8
2.8
0.9948 0.995 0.9952 0.9954 0.9954 0.9956 0.9958 0.9958 0.996 0.9962
2.9
2.92.9
2.9
0.9962 0.9964 0.9964 0.9966 0.9968 0.9968 0.997 0.997 0.9972 0.9972
3
33
3
0.9974 0.9974 0.9974 0.9976 0.9976 0.9978 0.9978 0.9978 0.998 0.998
