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Group Title: Research report
Title: Projection techniques for the non-statistically inclined
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082801/00001
 Material Information
Title: Projection techniques for the non-statistically inclined
Alternate Title: Research report 113 ; Florida State Department of Education
Physical Description: 22p. : ; 28cm.
Language: English
Creator: Florida -- Dept. of Education
Publisher: Florida Dept. of Education
Place of Publication: Tallahassee, Fla.
Publication Date: September, 1974
Copyright Date: 1974
 Subjects
Subject: Educational statistics   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
statistics   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: prepared by the Research Information and Surveys Section of the Bureau of Research and Information, Division of Elementary and Secondary Education.
 Record Information
Bibliographic ID: UF00082801
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 12627417

Table of Contents
    Front Cover
        Front page 1
        Front page 2
    Main
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
    Back Cover
        Page 23
        Page 24
Full Text




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DEPARTMENT OF EDUCATION
Tallahassee, Florida
RALPH D. TURLINGTON, COMMISSIONER


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SEPTEMBER 1974


g OF F. LIBRARY


PROJECTION

TECHNIQUES

for the

NON-STATISTICALLY

INCLINED


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State of Florida
Department of Education
Tallahassee, Florida
Ralph D. Turlington, Commissioner


Research Report 113 is a new concept in the report series and is designed to
provide districts and community colleges with methods for extrapolating base
line data. A companion report, Research Report 114 will provide the historical
data, where available, to facilitate the projections for each.

This report was designed and prepared by the Research Information and Surveys
Section of the Bureau of Research and Information, Division of Elementary and
Secondary Education, Department of Education. Inquiries regarding the Research
Report should be addressed to James A. Kemp, Educational Consultant, Research
Information and Surveys, 409 Knott Building, Tallahassee, Florida 32304 (450)








PROJECTION TECHNIQUES FOR THE
NON-STATISTICALLY INCLINED



I. Introduction

The value of interpolating the future of some element in the universe based

on the assessment of past and existing conditions, is obvious. Too often, however,

the potential for insight into a problem is not attained due, in part, to a lack of

understanding of basic projection techniques.

Three serious misconceptions regarding projection techniques are prevalent.

First, it should not be assumed that all methods are difficult. Although there

are many which are best handled by a computer, several require little more than

paper and pencil, and the rudiments of basic algebra. Some of the more useful of

these methods will be delineated later.

Second, it should not be assumed that because some standard technique has

been utilized, that all or any conclusions derived therefrom will be infallable.

Projections are merely estimates and as such can never be more accurate than the

data from which they were obtained. Environmental conditions impinging upon the

variable to be forecast can significantly alter the degree of accuracy. The most

accurate predictions occur when these outside conditions vary little from the

expected or the norm over a selected period of time.

Third, it should not be assumed that all methods of projection will generate

identical information concerning a specified event, even though all raw data may

have been identical. These discrepancies may be linked to the degree of rigor-

ousness of the technique used. In general, the more rigorous the projection

technique, the higher the probability that the resultant information will be less

contaminated.








The simple methods of projections outlined in this report, although not

difficult to compute, are not without merit. They are calculated quite rapidly

and under fairly stable conditions serve quite adequately.

II. Time Series

Introductory discussions about projection techniques must also address time

series upon which data the computations will be made. A time series is a repre-

sentation of some variable over any given length of time. When this variable is

represented statistically, its analysis is possible.

In general, there are four basic patterns which influence time series:

(1) long-term or basic trends, (2) seasonal fluctuations, (3) cyclical variations,

and (4) irregular fluctuations. The characteristics of each of these must be

inspected in order to understand the nature of possible discrepancies.

Long-term or basic trends involve relatively lengthy periods of time

relative to the duration of the phenomenon under study. Such statistical data

plotted on a graph would reveal a comparatively smooth pattern with no sudden

reversals or changes. Depending upon the type of graph used, the trend line may

be relatively straight or may gradually curve. In projecting variables based on

long-term trends it is assumed that the environmental elements which effect changes

in the specified variable will remain stable.

Seasonal fluctuations are controlled by two primary factors: climatic vari-

ations and local customs. That climatic fluctuations influence trends is easily

comprehendible. The latter factor is less obvious. Customs vary from nation to

nation, and from region to region. Included in the term "customs" would be holi-

days and religious influences, among others.

Cyclical variations are those which follow a definite pattern but which are

not bound by a calendar. Such cycles may be several years in duration. Ideally

these cycles should be of (near) identical length, but in reality external forces

often influence it, causing consecutive cycles of uneven length or magnitude. The









erratic length in cyclical variations is not as acute, however, when the variations

are viewed from the perspective of the much larger long-term trend. A number of

cyclical patterns of consequence have been identified such as the Julliard (10-year)

and the 37-month business cycles, and various weather cycles.

Irregular fluctuations are single or multiple, unique deviations from that

which has been identified as normal. Although usually isolated both in time and

in space from one another, a succession of unique elements can contribute signi-

ficantly to any trend, especially as the parameters controlling time and space

are increasingly restricted.

All four influences co-exist under most circumstances. In those situations

in which one or more irregular features dominate the contributions of the other

factors, the trend will become increasingly less reliable with the frequency and

magnitude of the fluctuations.

III. Techniques of Projection

Presented here are simple methods of predicting future values of a desired

variable. The description of these techniques have been kept as basic as possible.

In general, the techniques are presented in increasing order of difficulty.

Freehand Method. Like the other methods described below, this technique is

applicable only in comparing one variable with one other (i.e., it is two dimen-

sional). Data must first be arranged in some specified order, e.g. chronologically.

Next, this must be plotted on a graph and the consecutive points connected by

straight lines. A smooth curve may then be drawn along that imaginary line which

the eye perceives as fitting the data the best. (See figures 1 & 2.) One definite

advantage of the freehand method is that the line of interpolation may be a curve;

the other methods to be outlined will necessarily be straight line methods. The

extention of the curve past the last data point represents future predicted

values.






















Florida
Population
in Millions


9 /


8 -


7


6 ------ -


5


4


3 Trend Line--


2
1 Raw Data




1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990

FIGURE 1. FREEHAND METHOD


























12th Grade
Graduates in
Lime County


Trend L


Lne





-^^


-- Basic D


-7-


FIGURE 2. FREEHAND METHOD


1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976









Semi-Average. This method of projecting a trend involves basic mathematics.

It is extremely fast to calculate and is quite satisfactory when it has been

determined that the trend is linear.

The original data must first be arranged in some specified order and then

plotted on a graph, consecutive points connected by straight lines. The trend

period (horizonal ordinate) is divided into two equal parts and the arithmetic

mean value of the variable (verticle ordinate) is calculated for each. Any

extremely divergent values of the variable may be omitted from this computation.

This trend line will be more representative of the long-term trend than it would

have had the erratic data been included. The two average values are then plotted

at the midpoints of each period (see "X's" in fig. 3 at coordinates 1942,800

and 1962k, 1600 ) and a straight line is drawn between them and extending to

either side. The line extending to the right extrapolates the predicted,

future values of the variable. (See Figure 3.)

Example. Assume that it is desirable to predict the number of new residents in a

particular county.

Step 1. Arrange the data chronologically.

Number of
Year New Residents
1935 400
1940 900
1945 1100
1950 2500
1955 1200
1960 1500
1965 1800
1970 1900

Step 2. Plot the data on a graph and interconnect the points by short straight

lines (Figure 3).

Step 3. Divide the data into two equal chronological periods. Period 1: 1935,

1940, 1945, 1950; Period 2: 1955, 1960, 1965, 1970.


-6-









Step 4. Determine the average of the variable for each of the two periods,

eliminating from the computations any data which is extremely high or
400 + 900 + 1100
extremely low. Period 1: 3 = 800,
1200 + 1500 + 1800 + 1900
Period 2: 4 = 1600.

Step 5. Plot the two averages at the midpoint of each half. Point 1: (1942k,

800); Point 2: (1962, 1600).

Step 6. Draw a straight line between these two points and extending to either

side. This is the trend line. The extension of the trend line beyond

the last data point gives the predicted values of the variable.

Average of Period. This method is very similar to the last, the main

difference being the number of periods to be averaged to establish the trend.

The Average of Period method is slightly more sensitive than the semi-average

method, but like the semi-average is useful only for linear trends. (See Figure 4;

again computed averages for each period are marked by "X's").

As with the semi-average, this method of extropolation has little to

recommend it over the free-hand method.

Example. Assume that it is desirable to predict the number of new residents in

a particular county. (Compare this method with the semi-average,

above.)

Step 1. Arrange the raw data chronologically.
Number of
Year New Residents

1935 400
1938 900
1941 800
1944 900
1947 1200
1950 2500
1953 1500
1956 1300
1959 1500
1962 1500
1965 1800
1968 1800









Step 2. Plot the data on a graph, connecting each point with the next by a

straight line.

Step 3. Divide the data into several periods of equal duration, e.g., 9 years.

Period 1: 1935, 1938, 1941; Period 2: 1944, 1947, 1950; Period 3:

1953, 1958, 1959; Period 4: 1962, 1965, 1968. (Note that each period

begins 1' years before the first date given and extends 1 years

beyond the last date given).

Step 4. Compute the mean value of the variable for each period eliminating any
400 + 900 + 800
extremely high or extremely low value. Period 1: 3 = 700;
900 + 1200 2500 1500 + 1300 + 1500
Period 2: 3 = 1533; Period 3: 3 = 1433;
1500 + 1800 + 1800
Period 4: 3 = 1700.

Step 5. Plot each average at the midpoint of the period. Point 1: 1938,

Period 2: 1947; Period 3: 1956; Period 4: 1965.

Step 6. Connect each point by a short straight line. Use the straight line

between the last two averages (i.e., the last two "X's" in Figure 4)

as the line of extrapolation for future values.


-8-

























New
Residents


3000






2000






1000


1935 1940 1945 1950 1955 1960 1965 1970 1975 1980

FIGURE 3. SEMI-AVERAGE


-9-

























erio 1 I Peio ro IPro


Raw
Data -


Line


I i


Ln co H- 4 r- 0 ko r N LA co
m mA m Ln w w
I HR H 4 H HEA H H O-F PH R
FIGURE 4. AVERAGE OF PERIOD


-10-


Trend


New
Residents


3000





2000





1000


Ln 0o
H -


_ ___ ___


I Period 1I Period 4


|Period 3 I Period 4


.rrl








Moving Average. Like the semi-average and average of period, this method

utilizes a series of averages to establish a trend and to extrapolate future

values of a given variable. However, unlike the previous two methods, the Moving

Average exmploys overlapping periods and averages. This allows the trend to be

more sensitive to change.

In this method, data is again arranged in a specified order and plotted on a

graph. The total duration of the variable being studied must then be divided

into smaller components which will be grouped into overlapping sets and averaged.

For example, assume it is discovered that annual enrollment is increasing. This

data is then graphically plotted. The components are years and the chosen
-1- -2- -3-
Fiscal Year Increase in Enrollment 3 Year Average

1960 55
1961 50 -
1962 57 54.0
1963 50 52.3
1964 62 56.3
1965 75 62.3
1966 100 79.0
1967 140 105.0
1968 97 112.3
1969 90 109.0
1970 91 92.7
1971 87 89.3
1972 85 87.7
1973 88 86.7

set is three years. (See figure 5.)

Obtain the average for each overlapping, three-year period (See Column 3, above),

Plot this data at the midpoint of each period and connect the points by short

straight lines. To determine future values extend the last short line beyond the

period of average.


-11-
































Increase
in
Enrollment


150






100






50


0 9-4 N M "r Ln %,o t- o M 0 N" %D to %D ko wo %0 t-t'^ o ^ i* ^ r
oN C 0 0 O l t00 M)M
SH U H r- H HI H H H H

FIGURE 5. MOVING AVERAGE


-12-







Least Squares Method. This method may be used for both straight and curved

trends. It also forms the basis for linear regression. The least squares technique

is a method for fitting a line so that the sum of the squares of the deviations of

the variable above and below the line will be a minimum. The general process for

least squares is outlined below. A detailed explanation of each step is found in

the example.

Data must first be arranged in some specified order, e.g., chronologically.

(See Columns 1 & 2, page 16.) Compute the mean of the variable (See Column 2).

Next the deviations (or differences) from the midpoint are determined. In this

instance the midpoint is a year since time is the independent variable and the

variable to be predicted depends upon the passage of time. (Column 3, page 16).

Square the deviations (Column 4, page 16). Multiply the variables in Column 2

by the deviations in Column 3. Obtain the totals of the squared deviations and

the variable multiplied by the deviations. Divide the second total by the first.

This number gives the amount by which the variable in Column 2 increases on

the average from year to year. The graphic ordinate of the dependent variable

(Column 6, page 16) is computed by adding to the mean of Column 2 (for each year),

the product of the deviation (for each year) and the average annual increment.

See Figure 6 for a graphic representation.

Example. Assume that it is desirable to predict the total number of blind students

in the district.

Step 1. Collect data for the last few years showing the total number of blind

students in the district in each year. Data for at least four (4) years

should be used.

Step 2. Arrange this data chronologically.


-13-







Number of
Year Blind Students (Variable)

1960 16
1961 40
1962 30
1963 47
1964 55
1965 23
1966 41
1967 69
1968 60
1969 73

Step 3. If the number of years for which you have collected data is even, leave

a space between the middle two years and insert a small "dash" in the year

column and the variable column. Since the example includes a ten-year

period 1960-1969, these "dash" marks are inserted between 1964 and 1965.

Number of
Year Blind Students

1964 55

1965 23

Step 4. Add the number of blind students for each year (16 + 40 + 30 + 47 + 55

+ 23 + 41 + 69 + 60 + 73 = 454) and divide this total by the number of
454
years in the sample ( 10 45.4). This gives the average number of blind

students in the district over the ten-year period.

Step 5. Determine the middle of the time period of the sample. If the number of

years in the sample is even, then this point will fall between the middle

two years (1964 and 1965). If the number of years in the sample is odd,

then the middle year would be chosen.

Step 6. The deviation is the distance in time each year is from the "middle". In

this example this middle lies between two years, so each year will deviate

by some number plus or minus .5. (This example is true for any sample

containing an even number of years. If the example had an odd number of

years, then each deviation would be a whole number.) The deviations

of the years prior to the midpoint are proceeded by a minus sign, while

those following the midpoint are proceeded by a plus sign.
-14-






Step 7. Make a third column labeled "deviation". Place an "0" at the midpoint and

insert the deviation for all other years.

(Variable)
Number of
Year Blind Students Deviation

1960 16 -4.5
1961 40 -3.5
1962 30 -2.6
1963 47 -1.5
1964 55 .5
-- 0
1965 23 + .5
1966 41 +1.5
1967 69 +2.5
1968 60 +3.5
1969 73 +4.5

Step 8. Square each deviation in column 3 and enter these numbers in a fourth

column. (Variable)
Number of Squared
Year Blind Students Deviation Deviation

1960 16 -4.5 20.25
1961 40 -3.5 12.25
1962 30 -2.5 6.25
1963 47 -1.5 2.25
1964 55 .5 .25
0 0
1965 23 + .5 .25
1966 41 +1.5 2.25
1967 69 +2.5 6.25
1968 60 +3.5 12.25
1969 73 +4.5 20.25

Step 9. Add the "squared deviations column". (20.25 + 12.25 + 6.25 + 2.25 + .25

+ .25 + 2.25 + 6.25 + 12.25 + 20.25 O 82.50).

Step 10. For each year multiply the number of blind students in the district by the

deviation. Enter the answer in a new column.


-15-








Year

1960
1961
1962
1963
1964

1965
1966
1967
1968
1969





Step 11.





Step 12.














-1-

Year

1960
1961
1962
1963
1964

1965
1966
1967
1968
1969


(Variable)
Number of Squared
Blind Students Deviation Deviation Col. 2 x Col. 3

16 -4.5 20.25 72.0
40 -3.5 12.25 -140.0
30 -2.5 6.25 75.0
47 -1.5 2.25 70.0
55 .5 .25 27.0
0 0 0
23 + .5 .25 + 11.5
41 +1.5 2.25 + 61.5
69 +2.5 6,25 +172.5
60 +3.5 12,25 +210.0
73 +4.5 20.25 +328.5



73 (blind students) x 4.5 (deviation) = 328.5

Total all values obtained in Step 10 (column 5). Divide this number by
+399.0
that obtained in Step 9: 82.5 = 4.35. The value 4.35 is the average

annual increment of the variable.

Label the next column "graphic ordinates". When plotting the information

on a graph, the data in this column along with that in column 1 will mark

the points through which the trend line will pass. To obtain the values

in this column, for each year in this example, multiply the increment

(Step 11) by the deviation and add this product to the average number of

blind students (Step 4). For the year 1960 we would have: (4.35) (-4.5) +

45.4 = 25.82.

-2- -3- -4- -5- -6-
Squared Graphic
Variable Deviation Deviation Col.2 x Col.3 Ordinates

16 -4.5 20.25 72.0 25.82
40 -3.5 12.25 -140.0 30.18
30 -2.5 6.25 75.0 34.52
47 -1.5 2.25 70.5 38.88
55 .5 .25 27.5 43.22
0 0 0 45.40
23 + .5 .25 + 11.5 47.58
41 +1.5 2.25 + 61.5 51.93
69 +2.5 6.25 +172.5 56.28
60 +3.5 12.25 +210.0 60.63
73 +4.5 20.25 +328.5 64.98


MEAN 10Fi4 = 45.4

INCREMENT =


82.50


+399.0


399.0 = 4.35
82.5
-Ir-









Step 13. Plot the raw data on a graph and connect the points by solid, straight

lines. Enter the coordinates obtained in Steps 1 through 12 (column 1

and 6) on the graph and connect all these points by a broken line. This

line should be perfectly straight. If this broken line is extended

beyond the last data point, it then represents the line of predicted

values.







8 0 I. .. T........- ,


Blind
Students


M V MstU 10 O 0)
I to LE w kS o U E
.0 '0 M0 10 M (O010
Hr-1 H -4 H -4 H 4 H H-

FIGURE 6. LEAST SQUARES


0 H C


-17-


o0 H
0D %D W.0








Ratio Method. This method is widely used, but often is inferior to the last

method to be outlined in this paper, the Cohort Survival Technique. Its utility,

however, is that it allows for a very rapid calculation of an approximate future

value of a given variable.

In its most basic form a predicted value of a variable may be calculated by

dividing past values of the same variable by the total population from which it

was taken or by some other variable which has been shown to correlate very highly

and multiplying this by the total population (or related variable) of future years.

For example, assume that it is desirable to know the number of new students

a district can anticipate in the fall. Past records have shown that for every 100

new residential telephones installed between May and August 15th, 34 children will

enter the public school system in the fall. The ratio method assumes that this

relationship will continue unchanged. Therefore, if the local telephone company

records show that 275 new residential connections have been made, the estimated
34
number of new students would be: 100 x 275 = 93.5.

In problems dealing with the school population as a fraction of the age

pool, each age would be weighted. Although this makes the estimate more accurate,

it increases the complexity of the calculations to the point that this method has

nothing to offer that the Cohort-Survival Technique cannot offer more accurately.

Cohort-Survival Techniques. This group of closely related methods is based

upon the extent to which a particular phenomenon or groups of individuals can

survive through a sequence of pre-determined steps (e.g., grades 1, 2, 3, etc.).

This method, as opposed to several of the previous ones, does not lend itself to

graphic prediction, but rather is a succession of mathematical ratios.

The easiest way to explain the method is through an example. Assume that

it is desirable to predict the future public school average daily membership by

grade in Lime County.


-18-







Step 1. Obtain the birth statistics for the proceeding 10 years. Obtain the

average daily membership statistics for the current and proceeding five

years for grades 1 12 and arrange this in chronological order.

BIRTH DATA

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972


253 247 229 179 204 189 201 163 216 181 219


ADM FOR GRADES 1 12

Year 67-68 68-69 69-70 70-71 71-72 72-73


Grade
1 252 268 238 195 149 173
2 230 226 256 196 219 174
3 278 239 224 224 179 223
4 250 266 239 196 227 186
5 279 270 263 197 197 193
6 207 249 260 239 204 203
7 246 195 267 246 230 217
8 260 245 196 225 233 236
9 192 243 227 160 216 250
10 172 166 217 188 157 180
11 175 167 151 169 176 141
12 152 156 146 131 91 146

Total
K-12 2693 2690 2684 2366 2278 2322

Step 2. To calculate the survival ratio for first grade, total the number of resident

births for the five-year period 1962-66. Now find the total ADM for first

grade from 1968-1969 through 1972-1973. These are the students who were

enrolled in first grade six years later. Divide the total number of first
1023
grade students by the total number of births: 1112 = .92. The figure .92

is the average survival ratio of resident births to 1st graders.

Step 3. To estimate the future enrollment in first grade, multiply the number of

resident births for a given year by .92. This will give the approximated

first grade enrollment six years later.


-19-








BIRTH DATA

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972

253 247 229 179 204 189 201 163 216 181 219



ADM FOR GRADES 1 12

67-68 68-69 69-70 70-71 71-72 72-73 7 -74 74-75 75-76 76-77 77-78 78-79




SURVIVAL TIO
Known Predicted
Grade252 268 238 195 149 173 .92 174 185 150 199 167 201



201 (births in 1968) x .92 (survival ratio) = 185

(predicted 1st grades in 1974-75).


Step 4. To calculate the survival ratio for any two consecutive grades, add the

ADM for 5 consecutive years (e.g., 1967-68 through 1971-72, inclusive) for

the lower of the two grades. Add the ADM for 5 consecutive years for the

upper of the two consecutive grades beginning 1 year later (e.g., 1968-69

through 1972-73, inclusive). Divide the second total by the first. In
226 256 196
this example the survival ratio for second grade would be 253 + 268 + 238
219 174 1071
+ 195 + 149 = 1102 = .97.

Step 5. To estimate the future enrollment for any grade, multiply the survival ratio

for the grade by the number of students in the next lower grade one year

before.


-20-







ADM FOR GRADES 1-12

Year 67-68 68-69 69-70 70-71 71-72 72-73 73-74 74-75 75-76 76-77 77-78 78-79



SURVIVAL RATIO


238
256
224
239
263
260
267
196
227
217
151
146


195
196
224
196
197
239
246
225
160
188
169
131


149
219
179
227
197
204
230
233
216
157
176
91


173
174
223
186
193
203
217
236
250
180
141
146


.92 174 185 150 199
4.97 168 j169 180 146
.97 168 162 163 174
.97 217 164 158 159
.95 177 206 156 150
.96 185 169 198 149
1.00 202 184 169 197
.96 208 194 177 162
.95 223 197 183 167
.87 219 195 172 160
.89 161 195 174 154
.80 113 129 158 139
-X _-= --

194 (1st grades in 1973-74) x .97

= 169 (2nd grades in 1974-75).


167
193
141
169
151
144
149
189
153
146
143
123


201
162
187
137
161
145
144
142
179
134
130
115


(survival ratio)


In the Cohort-Survival method, errors appear to be cyclical which will

necessitate the yearly revision of the ratios. The following table gives the

complete data for the Lime County example. Note that in this projection the

survival ratio was computed to four decimal places and rounded to two (2) in this

table. This accounts for all discrepancies which may be encountered.


-21-


trade
1
2
3
4
5
6
7
8
9
10
11
12


252
230
278
250
279
207
246
260
192
172
175
152


268
226
239
266
270
249
195
245
243
166
187
156








COHORT SURVIVAL PROJECTION
Lime County

BIRTH DATA

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972


229


179


204


201


216


219


ADM FOR GRADES 1-12


7~-7 7475 5-76 76-7 7-78 78-79


YEAR

GRADE
1
2
3
4
5
6
TOTAL
1-6

7
8
9
TOTAL
7-9

10
11
12
TOTAL
10-12


149
219
179
227
197
204

1175

230
233
216

679

157
176
91

424


2278


173
174
223
186
193
203


Survival
Ratio
.92
.97
.97
.97
.95
.96


1152


217
236
250


1.00
.96
.95


174
168
168
217
177
185

1089

202
208
223


219
161
113


195
196
224
196
197
239

1247

246
225
160

631

188
169
131

488


2366


2498


252
230
278
250
279
207

1496

246
260
192

698

172
175
152

499


2693


185
169
162
164
206
169

1056

184
194
197

575

195
195
129

519


2150


1'50
180
163
158
156
198

1005

169
177
183

529

172
174
156

503


2036


199
146
174
159
150
149

977

197
162
167

526

160
154
139

454


1956


167
193
141
169
151
144

965

149
189
153

490

146
143
123

412


1868


201
162
187
137
161
145

993

144
142
179

465

134
130
115

379


1836


2425 2472 2363 2270


268
226
239
266
270
249

1518

195
245
243

683

166
167
156

489


2690


TOTAL
K-12


238
256
224
239
263
260

1480

267
196
227

690

217
151
146

514


2684


180
141
146

467


2322


2215


r-'7 r- 0 r- ;r_ a rQ-'7n '7 -7"> 72-73 73-4 74-75 75-76 7 -77 77-78 78-7


73-71


2195 2089 2029


2869 2885 2880



















































































































































































































b














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