The Normal Approximation to Data

Many (but probably not most) histograms of data can be approximated by a Normal Curve if the data values are converted to Standard Units, in the sense that the area under the histogram over various regions is about the same as the area of the normal curve over the same regions. The following applet shows the histogram of 100 measurements of the acceleration of gravity at Piņon Flat Observatory.

The scroll bars let you select different regions of the histogram; the area under the histogram and under the corresponding part of the normal curve are shown at the bottom of the applet.

The following is a table of 100 observations of g, the acceleration due to gravity, made at Piņon Flat Observatory in 1989 (day 229, between 5:29:52pm and 5:48:08pm). The base g value is 9.792838 meters/s**2. The table lists the deviations of the measurements from that reference value. The experimental apparatus is pretty slick: it uses a laser and an accurate time reference to determine the distance a mirrored corner of a cube falls in a vacuum chamber as a function of time. These measurements were made by Glen Sasagawa and Mark Zumberge.

-.00000001 .00000002 -.00000074 -.00000044 -.00000019

-.00000060 -.00000106 -.00000083 -.00000086 -.00000079

-.00000024 -.00000012 .00000014 -.00000074 -.00000050

.00000041 .00000014 .00000072 -.00000015 -.00000107

-.00000071 -.00000128 -.00000035 -.00000120 .00000018

-.00000101 -.00000019 -.00000067 .00000024 -.00000027

-.00000106 -.00000000 -.00000150 .00000000 .00000014

-.00000132 -.00000014 .00000018 -.00000014 .00000051

-.00000037 -.00000034 -.00000059 -.00000113 .00000029

-.00000083 -.00000038 -.00000108 -.00000061 -.00000016

.00000045 -.00000069 -.00000034 -.00000004 -.00000071

-.00000132 -.00000089 -.00000055 -.00000024 .00000019

-.00000099 .00000155 .00000001 -.00000012 -.00000067

.00000014 -.00000112 -.00000012 -.00000016 -.00000060

-.00000043 -.00000062 -.00000019 -.00000080 -.00000019

-.00000122 -.00000049 .00000029 -.00000101 -.00000074

.00000007 -.00000018 -.00000105 -.00000121 -.00000027

-.00000016 -.00000048 -.00000029 -.00000065 -.00000048

-.00000080 -.00000152 -.00000052 -.00000026 -.00000087

-.00000107 -.00000054 -.00000054 -.00000047 -.00000106

Why take so many measurements?

• Fluctuations tend to average out to some extent, so you can get more accurate results by averaging the measurements
• Quality control: if the fluctuations get larger, or a measurement is "unusual," something might be going wrong with the instrument or its set-up.
• The variability of the repeated measurements gives you an estimate of the uncertainty of a single measurement, or of the average of the measurements, via the SD.