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In the figure a there are three kinds of stationary points:
♦ Point A is a maximum point.
♦ Point B is a minimum point.
♦ Point C is a stationary point of inflection.
At a point of inflection the tangent changes from one side of the curve to the other. In the figure b point D is also a point of inflection, but since the gradient at that point is not zero, it is not a stationary point.
A stationary point can also be explained as a point where the function is neither increasing nor decreasing, hence stationary! A turning point on a curve is either a maximum point or a minimum point. It is a point where a function changes from increasing to decreasing, or vice versa. A critical point is also known as a stationary point.
References: Concave; Decreasing Function; Gradient; Gradient of a Curve; Inflection, Point of; Maximum Value; Slope; Tangent.
This is a branch of mathematics that is about the collection, display, and analysis of numerical data in order to draw conclusions or make predictions. For information on the various terms in statistics, see the respective entries.
References: Arrangements, Average, Decile, Factorial, Frequency Distribution, Mean, Median of a Set of Data, Mode, Normal Distribution, Percentiles, Quartiles, Range, Scatter Diagram, Selection, Spread, Standard Deviation, Variance.
STEM AND LEAF GRAPH
This graph, which is used in statistics, is really a histogram in which each item of the data is shown in the graph. The stem and leaf graph has the appearance of a histogram, but none of the original data are lost. The examples illustrate the concept of a stem and leaf graph.
Example 1. Nathan collected data about the weights, in kilograms, of the students in his class of 25 students. The weights were recorded to the nearest kilogram. Draw a stem and leaf graph of the data, which are listed here:
39, 42, 45, 46, 46, 49, 51, 53, 53, 55, 57, 57, 58, 58, 59, 60, 61, 62, 63, 64, 64, 67, 68,71, 75
Solution. Each item of data is made up of tens and units. For example, the first item of data is 39, which is three tens and nine units. The tens numbers form the “stem of the plant” and the units numbers form the “leaves of the plant.” The result is shown in figure a and is called a stem and leaf graph. It is customary to write only one digit
STEM AND LEAF GRAPH 429
in the stem, so that 3 stands for 30, 4 stands for 40, and so on. The two graphs are drawn side by side. Also included is a histogram for comparison.
Stem and leaf graph Stem and leaf graph
25 669 133577889
0 1 2344 78
25 669 133577889
0 1 2344 78
30 40 50 60 70 80
Weights of students, kg
Back-to-back stem and leaf graphs are used to compare two sets of data, as shown in the next example.
Example 2. The back-to-back stem and leaf graph in figure b compares the results of the 1972 Olympic final in the 400 meters hurdles with the 1992 final 20 years later. There were eight finalists in each of the two events. The times are given in seconds, to 2 dp. The time for the first-place runner in 1992 (the fastest time in the graph) is 46.78 seconds; the time for the second-place runner in 1992 is 47.66 seconds; that for the first-place runner in 1972 is 47.82 seconds; etc.
Back to Back Stem and Leaf Graph
82 47 66 82
64 52 21 48 13 63 63 86
66 66 65 49 26
As a matter of interest, if all 16 athletes ran in the same race and finished the race in their Olympic times, then their places would be as shown in the table, where 92 and 72 indicate runners from those respective years. The best 1972 runner would have tied for a bronze medal in the 1992 final.
1st 2nd 3rd 5 th 6 th 7th 8th 10th 11th 12th 13th 14 th 15th 16th
92 92 92 92 72 72 92 72 72 92 72 72 72 72
The back-to-back stem and leaf graph shows that athletic performance in the 400 meters hurdles did not improve by a great amount in those 20 years. The mean time for the 1972 race was 49.05 seconds and that for the 1992 race was 48.22 seconds. The percentage improvement is
x 100 = 1.7% (to 1 dp)
49.05 v 1'
References: Histogram, Mean, Percentage, Statistics.
Reference: Acute Angle.
References: Cartesian Coordinates, Gradient-intercept Form, Graphs.
SUBJECT OF A FORMULA
Reference: Changing the Subject of a Formula.
Substitution usually means replacing a variable in a formula or equation by a number that represents the variable, or by another variable. Various examples of substitution are described in the entries given in the references. One type is explained here, which is replacement by numbers of the variables in algebraic terms.
Example. If x = — 2, y = 3, and z = 0, find the values of each of the expressions
(a) 2x — 3y and (b) (x2 + y2)jz2.
Solution. For (a), write 2x — 3y = 2 x (—2) — 3 x 3 Substituting —2 for x, in brackets, and 3 for y = -4-9 = -13 For (b), write
x2 + y2