HP 15c Manual

							 Section 4: Statistics Functions 51 
 
X NITROGEN APPLIED 0.00 20.00 40.00 60.00 80.00 (kg per hectare *), x 
Y 
GRAIN YIELD 
4.63 4.78 6.61 7.21 7.78 (metric tons per 
hectare), y 
*A hectare equals 2.47 acres. 
Keystrokes Display  
´ CLEAR ∑ 0.0000 Clears statistical storage registers (R2 through R7 and the stack). 
´ • 2 0.00 Limits display to two decimal places, like the data. 
4.63 v 4.63  
0 z 1.00 First data point. 
4.78 v 4.78  
20 z 2.00 Second data point. 
6.61v 6.16  
40 z 3.00 Third data point. 
7.21 v 7.21  
60 z 4.00 Fourth data point. 
7.78 v 7.78  
80 z 5.00 Fifth data point. 
l 3 200.00 Sum of x-values, Σx (kg of nitrogen). 
l 4 12.000.00 Sum of squares of x-values, Σx2. 
l 5 31.01 Sum of y-values, Σy (grain yield). 
l 6 200.49 Sum of squares of y-values, Σy2. 
l 7 1,415.00 Sum of products of x- and  y-values, Σxy.  
						

							52 Section 4: Statistics Functions 
 
Correcting Accumulated Statistics 
If  you  discover  that  you  have  entered  data  incorrectly,  the  accumulated 
statistics  can  be  easily  corrected.  Even  if  only  one  value  of  an  (x,  y)  data 
pair is incorrect, you must delete and re-enter both values. 
1. Key the incorrect data pair into the Y- and X-register. 
2. Press |w to delete the incorrect data. 
3. Key in the correct values for x and y. 
4. Press z. 
Alternatively,  if  the  incorrect  data  point  or  pair  is  the  most  recent  one 
entered  and z has  been  pressed,  you  can  press |K |w to 
remove the incorrect data.* 
Example:  After keying in  the preceding data. Farmer realizes  he  misread a 
smeared  figure  in  his  lab  book.  The  second y-value  should  have  been  5.78 
instead of 4.78. Correct the data input. 
Keystrokes Display  
4.78 
v 
4.78 Keys in the data pair we want to replace 
and deletes the accompanying statistics. 
The n-value drops to four. 20 |w 4.00 
5.78 
v 
5.78 Keys in and accumulates the replacement 
data pair. 
20 z 5.00 The n -value is back to five. 
We will use these statistics in the rest of the examples in this section. 
                                                           * Note that these methods of data deletion will  not delete any rounding errors that  may  have been  generated in the statistics registers. This difference will not be serious unless the erroneous pair has a  magnitude that is enormous compared with the correct pair, in such a case, it would be wise to start over!  
						

							 Section 4: Statistics Functions 53 
 
Mean 
The ’ function  computes  the  arithmetic  mean  (average)  of  the x-and y-
values  using  the  formulas  shown  in  appendix  A  and  the  statistics 
accumulated in the relevant registers. When you press |’ the contents 
of the stack lift (two registers if stack lift is enabled, one if not); the mean of 
x ( x)  is  copied into  the  X-register  as  the  mean  of y ( y)  is  copied 
simultaneously into the Y-register. Press ® to view  y. 
Example: From  the  corrected  statistics  data  we  have  already  entered  and 
accumulated,  calculate  the  average  fertilizer  application,  x. and  average 
grain yield  y, for the entire range. 
Keystrokes Display  
|’ 40.00 Average kg of nitrogen,  x, for all cases. 
® 6.40 Average tons of rice,  y, for all cases. 
Standard Deviation 
Pressing |S computes  the  standard  deviation  of  the  accumulated 
statistics  data.  The  formulas  used  to  compute sx,  the  standard  deviation  of 
the accumulated x-values, and sy, the  standard deviation of the accumulated 
y-values, are given in appendix A. 
This  function  gives  an  estimate  of  the  population standard  deviation from 
the  sample  data,  and  is  therefore  termed  the sample standard  deviation.* 
When you press |S, the contents of the stack registers are lifted (twice 
if stack lift is enabled, once if not); sx is placed into the  X-register and sy is 
placed into the Y-register. Press ® to view sy. 
                                                           * When  your  data  constitutes  not  just  a  sample  of  a  population  but  all  of  the  population,  the  standard deviation  of  the  data  is  the  true  population  standard  deviation  (denoted ).  The  formula  for  the  true 
population  standard  deviation  differs  by  a  factor  of   from  the  formula  used  for  the S 
function.  The difference between the  values is small  for  large  n, and  for  most applications can be  ignored. But  if  you  want  to  calculate  the  exact  value  of  the population  standard  deviation  for  an  entire  population, you can easily do so: simply add, using z, the mean ( x) of the data to the data before pressing |S. The result will be the population standard deviation. (If you subsequently correct any of your accumulated data values, remember to delete the first mean value and add the corrected one.) nn/)1(  
						

							54 Section 4: Statistics Functions 
 
Example: Calculate  the  standard  deviation  about  the  mean  calculated 
above. 
Keystrokes Display  
|S 31.62 Standard deviation about the  mean  nitrogen 
application,  x. 
® 1.24 Standard  deviation  about  the  mean  grain 
yield,  y. 
Linear Regression 
Linear  regression  is  a  statistical  method  for  finding a straight  line  that  best 
fits  a  set  of  two or more  data  pairs,  thus  providing  a  relationship  between 
two or more data pairs, thus providing a relationship between two variables. 
By  the  method  of  least squares, ´L will  calculate  the  slope, A, and y-
intercept, B, of the linear equation: 
y=Ax+B 
1. Accumulate the statistics of your data using the z key. 
2. Press ´L. The y-intercept, B, appears in the  display (X-
register).  The  slope, A,  is copied  simultaneously  into  the  Y-
register. 
3. Press ® to view A. (As is the case with the functions ’ 
and S, L causes  the  stack  to  lift  two  registers  if  its 
enabled, one if not). 
 
T t  y  y  
Z z  x  y  
Y y  A slope B y-intercept 
X x  B y-intercept A slope 
Keys:  ´L  ®   
The  slope  and  y-intercept  of  the  least  squares  line  of  the  accumulated  data 
are calculated using the equations shown in appendix A.  
						

							 Section 4: Statistics Functions 55 
 
Example: Find the y-intercept and slope of the linear approximation of the 
data and compare to the plotted data on the graph below. 
 
Keystrokes Display  
´L 4.86 y-intercept of the line. 
® 0.04 Slope of the line. 
Linear Estimation and Correlation Coefficient 
When  you press ´j the linear estimate, ŷ, is placed in the  X-register 
and  the correlation  coefficient,  r, is  placed  in  the  Y-register.  To  display r, 
press ®.   
						

							56 Section 4: Statistics Functions 
 
Linear  Estimation. With the  statistics accumulated, an estimated value  for 
y,  denoted ŷ,  can  be  calculated  by  keying  in  a  proposed  value  for x and 
pressing ´j. 
An Estimated value for x (denoted) can be calculated as follows: 
1. Press ´L. 
2. Key in the known y-value. 
3. Press ® - ® ÷. 
Correlation  Coefficient. Both  linear  regression  and  linear  estimation 
presume  that  the  relationship  between  the x and y data  values  can  be 
approximated by  a  linear  function.  The  correlation  coefficient, r, is a 
determination of how closely your data fit a straight line. The range is -1  r 
 1,  with -1 representing  a  perfectly  negative  correlation  and  +1 
representing a perfectly positive correlation. 
Note that  if  you  do  not  key  in  a  value  for x before  executing ´j, the 
number  previously  in  the  X-register  will  be  used  (usually  yielding  a 
meaningless value for ŷ). 
Example: What if 70 kg of nitrogen fertilizer were applied to the rice field? 
Predict  the  grain yield  based  on  Farmer’s  accumulated  statistics.  Because 
the  correlation  coefficient  is  automatically  included  in  the  calculation,  you 
can view  how closely the  data  fit a  straight line  by pressing ® after the 
y prediction appears in the display. 
 xˆ   
						

							 Section 4: Statistics Functions 57 
 
 
Keystrokes Display  
70 ´j 7.56 
 
Predicted grain yield in tons/hectare. 
® 0.99 The original data closely approximates a 
straight line. 
Other Applications 
Interpolation. Linear  interpolation  of  tabular  values,  such  as  in 
thermodynamics and statistics tables,  can be  carried out  very  simply on the 
HP-15C  by  using  the j function.  This  is  because  linear  interpolation  is 
linear  estimation:  two  consecutive  tabular  values  are  assumed  to  form  two 
points  on  a  line,  and  the  unknown  intermediate  value  is  assumed  to  fall  on 
that same line. 
Vector  Arithmetic. The  statistical  accumulation  functions  can  be  used  to 
perform  vector  addition  and  subtraction.  Polar  vector  coordinates  must  be 
converted  to  rectangular  coordinates  upon  entry  (θ, v, r ;, z). 
The results  are  recalled  from  R3 (Σx)  and  R5 (Σy)  (using l z)  and 
converted  back  to  polar  coordinates,  if  necessary.  Remember  that  for  polar 
coordinates  the  angle is  between -180°  and  180°  (or -π  and  π radians,  or -
200  and  200  grads). To  convert  to a positive  angle,  add  360  (or  2π  or  400) 
to the angle. 
For  the  second  vector  entered,  the  final  keystroke  will  be either z or 
w, depending on whether the two vectors should be added or subtracted. 
  
						

							 
58 
Section 5 
The Display 
and Continuous Memory 
Display Control 
The  HP-15C  has  three  display  formats – •, i, and ^  – that 
use a given number (0 through 9) to specify display format. The illustration 
below shows how the number 123,456 would be displayed specified to four 
places in each possible mode. 
´ • 4 : 123,456.0000 
´ i 4 : 1.2346    05 
´ ^ 4 : 123.46    03 
Owing to Continuous Memory, any change you make in the display format 
will be preserved until Continuous Memory is reset. 
The current display format takes effect when digit entry is terminated; until 
then, all digits you key in (up to 10) are displayed. 
Fixed Decimal Display 
• (fixed  decimal) format  displays  a  figure  with  the  number  of  decimal 
places you specify (up to nine, depending on the size of the integer portion.) 
Exponents  will  be  displayed if  the  number  is  too  small  or  too  large  for  the 
display. At ―power-up,‖ the HP-15C is in • 4 format. The key sequence 
is ´• n. 
Keystrokes Display  
123.4567895 123.4567895  
´• 4 123.4568  
´• 6 123.456790 Display is rounded to six decimal 
places. (Ten places are stored 
internally.) 
´• 4 123.4568 Usual • 4 display.  
						

							 Section 5: The Display and Continuous Memory 59 
 
 
Scientific Notation Display 
i (scientific) format  displays  a  number  in  scientific  notation.  The 
sequence ´i n specifies the  number of decimal  places to be  shown. 
Up  to  six  decimal  places  can  be shown  since  the  exponent display  takes 
three  spaces.  The  display  will  be  rounded  to  the  specified  number  of 
decimal  places;  however,  if  you  specify  more  decimal  places  than  the  six 
places the display can hold (that is, i 7, 8, or 9), rounding will occur in 
the undisplayed seventh, eighth, or ninth decimal place.* 
With the previous number still in the display: 
Keystrokes Display  
´i 6 1.234568  02 Rounds to and shows six 
decimal places. 
´i 8 1.234567  02 Rounds to eight decimal places, 
but displays only six. 
Engineering Notation Display 
^ (engineering) format  displays  numbers  in  an  engineering  notation 
format in a manner similar to i, except: 
 In  engineering  notation,  the  first  significant  digit  is  always  present  in 
the  display.  The  number  you  key  in  after ´^ specifies  the 
number of additional digits to which you want to round the display. 
 Engineering notation shows all exponents in multiples of three. 
Keystrokes Display  
.012345 0.012345  
´^ 
1 
12.       -03 Rounds  to  the  first  digit  after 
the leading digit. 
´^ 3 12.35     -03  
10 * 123.5     -03 Decimal  shifts  to  maintain 
multiple of three in exponent. 
´• 4 0.1235 Usual • 4 format. 
                                                           * Therefore, the display shows no distinction among i. 7, 8, and 9 unless the number rounded up is a 9, which carries a 1 over into the next higher decimal place.  
						

							60 Section 5: The Display and Continuous Memory 
 
Mantissa Display 
Regardless  of  the  display  format,  the  HP-15C  always  internally  holds  each 
number as a 10-digit mantissa and a two-digit exponent of 10. For example, 
π  is  always  represented  internally  as  3.141592654×1000,  regardless  of  what 
is in the display. 
When  you  want  to  view  the  full  10-digit  mantissa  of  a  number  in  the  X-
register,  press ´ CLEAR u.  To  keep  the  mantissa  in  the  display, 
hold the u key down. 
Keystrokes Display 
| $ 3.1416 
´ CLEAR 
u (hold) 3141592654 
Round-Off Error 
As  mentioned earlier, the  HP-15C holds every  value  to 10  digits  internally. 
It also rounds the  final result of every calculation to the 10th digit. Because 
the calculator can provide only a finite approximation for numbers such as  
or 2/3 (0.666…), a small error due to rounding can occur. This error can be 
increased in lengthy calculations, but  usually  is insignificant.  To accurately 
assess this effect for a given calculation requires numerical  analysis beyond 
our  scope  and  space  here!  Refer  to  the HP-15C  Advanced  Functions 
Handbook for a more detailed discussion. 
Special Displays 
Annunciators 
The  HP-15C  display  contains  eight  annunciators  that  indicate  the  status  of 
the  calculator  for  various  operations.  The  meaning  and  use  of  these 
annunciators is discussed on the following pages: 
 
* Low-power indication, page 62. 
USER User mode, pages 79 and 144. 
f and g Prefixes for alternate functions, pages 18-19. 
RAD and GRAD Trigonometric modes, page 26. 
C Complex mode, page 121. 
PRGM Program mode, page 66.