One of the tools used by environmentalists,

CALCULATING A DAILY NORMAL TEMPERATURE RANGE THAT REFLECTS DAILY TEMPERATURE VARIABILITY BY CHRISTOPHER HOLDER, RYAN BOYLES, PETER ROBINSON, SETHU RAMAN, AND GREG FISHEL By utilizing daily normal ranges, weathercasters and others could better communicate to the public the natural expected variability of day-to-day temperatures a variability not suggested by normals derived from smoothed trend lines. One of the tools used by environmentalists, meteorologists, and the media to gauge phenomena such as global or local warming or cooling trends is a particular area's normal temperature, produced by the National Climatic Data Center (NCDC). After quality control algorithms are run, the monthly normal temperatures are calculated as simple 30-yr monthly averages. Daily normals are AFFILIATIONS: HOLDER, BOYLES, ROBINSON, AND RAMAN State Climate Office of North Carolina, North Carolina State University, Raleigh, North Carolina; FISHEL WRAL-TV5, Raleigh, North Carolina CORRESPONDING AUTHOR: Sethu Raman, State Climate Office of North Carolina, North Carolina State University, Raleigh, NC 27695 E-mail: sethu_raman@ncsu.edu The abstract for this article can be found in this issue, following the table of contents. DOI: 10.1175/BAMS-87-6-769 In final form 10 January 2006 2006 American Meteorological Society then derived from the monthly normals using a cubicspline interpolation, a process that produces a smooth data curve (NCDC 2004). The final product, called "daily normal temperature," is a statistically sound number that can be used to examine day-to-day temperature trends. One drawback of climatological averages such as this lies in the fact that they are merely averages of extremes. Daily and weekly temperature variations can be dramatic, and rarely do temperature observations in most locations follow a smooth, steady trend for a significant period of time. Nearly every day will either be some amount cooler or warmer than this normal temperature, leading to impulsive conclusions that, for example, global warming is having an effect and that our weather is becoming more erratic. Most people do not take into account the day-to-day variability that is especially inherent in temperature. Thus, comparing any particular day's temperature observations with this normal temperature, as occurs often in media such as local news and weather broadcasts, can be misleading to many and lead to the following fundamental question: if we are so often AMERICAN METEOROLOGICAL SOCIETY JUNE 2006 BAFft 769

below or above normal, what, exactly, is "normal" (Lupo et al. 2003)? Therefore, giving the public a normal range of temperature for a particular date a span of temperatures that is derived from past observations and reflects day-to-day temperature variance may be more meaningful. The statistical techniques used to calculate this range are outlined in the "Procedure" section, results are presented in the "Observations" section, and final thoughts are presented in the "Conclusions" section. P R O C E D U R E. This study focuses on the 30-yr period from 1971 to 2000 at six locations that span North Carolina Raleigh-Durham, Charlotte, and Greensboro in the Piedmont; Asheville in the mountain region; Fayetteville in the coastal plain; and Wilmington on the coast. The 30-yr daily normal temperatures are calculated by the NCDC, and the actual daily temperatures for the various locations are obtained from NCDC (NCDC 2004). The standard deviation (a), which is the square root of the variance and gives a good indication of how much a particular parameter tends to vary across a dataset, is calculated for maximum temperature () and minimum temperature () on each calendar day (each 1 January, 2 January, etc.). To make this statistic more useful, however, a measure of probability can be derived from various multiples of o above and below the average if the o dataset is normally distributed, which is a statistical property of a dataset commonly called a bell curve because of its bell shape, or a Gaussian distribution. Other studies have found that the temperature data either 1) have a Gaussian distribution or 2) are close enough to Gaussian that the departures from the Gaussian curve are small enough to allow for an assumption of Gaussian distribution (Bingham 1961; Bruhn et al. 1980; Lupo et al. 2003). Here, the temperature variances are assumed to follow a bell curve. With that condition, one o above and below the mean, for example, contains about 68% of the dataset; that is, 68% of the 1 January maximum temperature values from the past 30 yr lie between (mean - a) and (mean + o). These areas can be determined using a z table, a basic tool found in most statistics textbooks or online. Here, (0.6745 a), henceforth called "our calculation," is used because it provides a 50% area. One can then say that, for example, 50% of the 1 January maximum temperatures in the last 30 yr fell within a particular range of values. The percentage choice is rather arbitrary. A higher percentage may be desirable at face value, but it would 770 I BATIS- JUNE 2006 require a larger range in temperature variability. The endpoints of the area under the curve must move further from the mean to encompass more data and provide a higher percentage. Eventually the reasonability of the size of the temperature range is sacrificed for a higher probability. This study uses 50% as a reference value to provide a probability that is understandable and intuitive to the common public user, but does not provide a temperature range that is so large that it loses value and significance. Other locations, such as the Midwest, may find that a smaller percentage is necessary because of a larger temperature range, but it is not clear at this point if doing so will create undesired uniformity issues regarding the definition and understandability of "range" from location to location. Given that the public seems comfortable with the usage of the normal temperature, this study uses the normal temperature, rather than the raw 30-yr daily average, as the mean of the distribution. Thus, our calculation is added to and subtracted from the NCDC daily normal temperature at the six locations in this study. OBSERVATIONS. The size of the ranges provided by these calculations seems reasonable (see Figs. 1,2) and helps to represent temperature variability inherent in a certain time of year. Graphs of our calculation for and of the 365 days of the year indeed show that temperature spreads tend to vary by season (Fig. la shows that of Raleigh-Durham). On average, winter varies about 4 times as much as summer, and winter varies about 5 times as much as summer (Table 1). The average largest 50% probability spread for among the six stations is 5.3 C. For the 10 days that have the largest spread, the median date is 8 February (Table 1). For, the average largest spread is 4.9 C and the 10-day median is 9 January. The smallest () spread is 1.4 C ( C), with a 10-day median of 5 August (26 July). The larger variance in winter is expected and can be attributed to the more dynamic weather systems and atmospheric flows that exist in the cooler months, when a greater latitudinal temperature gradient exists. Eleven-day moving-average trend lines reduce the magnitude of the daily temperature variations to better visualize how the time series of our calculations vary between and. The time period of eleven days is used because it is large enough to significantly reduce the magnitude of daily variance, while being small enough to capture significant trend changes on a small time scale. Ten days is not used because

even numbers are not symmetrical about a central point. With this smoother trend line, is more easily seen to vary more than throughout the year, except for the August- October period and for a portion of May, when the opposite is true. This pattern of versus is seen at all stations except for Wilmington, which shows more intertwining throughout the year (Fig. lb). At Wilmington, land-sea interactions add more complexity to the temperature variance signal. Basic issues, such as clear versus cloudy, moist versus dry, windy versus calm, etc., are key to values of and and are often directly related to topography, season, land usage, and geography. The lower limit of is governed, in part, by the dewpoint temperature. Dewpoint tends to vary less day to day than does, whose upper limit tends to be partially dictated by amount of sunshine, which is itself highly variant. During autumn, however, North Carolina tends to receive less rainfall, and clear skies are more dominant. With a tendency toward fewer clouds, tends to vary less and radiational cooling can be a prominent player in nighttime temperatures. However, wind is also a factor in the magnitude of radiational cooling, and winds and mesoscale systems tend to be highly variant in the spring and autumn months, which are times of transition. The on clear, calm nights can also vary significantly due to local topography and soil type, among other things. On a smaller time scale, there are significant dayto-day deviations in the time series of our calculation, as shown in Figs. la-b. Some daily perturbations are expected, but those calculated in this study are larger FIG. I. (a) Eleven-day running averages of the standard deviations ( F) of (red) and (blue) for Raleigh-Durham. Note the decrease in variance in warmer months, and that is more varied than at all times except autumn, (b) Same as (a), but for Wilmington. Note the intertwining of the graph lines, which is unique to Wilmington in this study. This is probably due to the complex land-sea interactions experienced at the coast. than anticipated. At certain times of the year, the signal of our calculation follows a nearly sinusoidal pattern. These daily fluctuations in the range may at face value confuse some people why, for example, is today's range 7 F when yesterday's was 5 F (degrees Fahrenheit is often used in presenting temperature data to the American public)? For example, Table 2 shows the and normal ranges at Greensboro for 1-10 January. Note that the range changes as much as 4.4 F from one day to the next, and the largest day-to-day difference in the normal range is 4.8 F for these 10 days. This variance markedly contrasts with the rather steady AMERICAN METEOROLOGICAL SOCIETY JUNE 2006 BAflS- I 771

(e.g., Lupo et al. 2003), but capturing observed daily variability is central to this study. Smoothing, such as through cubic splines and weighted or nonweighted moving averages, would minimize day-to-day variance in favor of reducing what appears to be noise in the signal of temperature variability. Given that the ranges are calculated from actual daily observations, these "noisy" features indeed represent daily temperature variability. As such, these data are not smoothed. This may warrant additional explanation by those, such as broadcast meteorologists, who may utilize these normal ranges. The amount of clarification that these users provide may vary depending on the audience and how in depth the users wish to go in explaining the product. One example of how the normal ranges could FIG. 2. Graph of N C D C - n o r m a l i z e d T m a x and T m i n for Raleigh-Durham, along be graphically presented w i t h t h e upper and lower limits of t h e T m a x and T m i n ranges, calculated by to the public is shown in (0.6745 x standard deviation) and added t o and subtracted f r o m t h e n o r m a l Fig. 3. One thermometer temperature. NCDC normal and data, which are also shown in Table 2. The variance in the normal and ranges as compared to the normal temperature curves can also be seen graphically in Fig. 2. Some precedent does exist for smoothing these raw data to reduce the magnitude of day-to-day variation. This smoothing would provide curves of our calculations that are as "clean" as the NCDC normal curves 1. M a x i m u m and m i n i m u m ranges of T m a x and T m i n ( F ) at t h e six locations in this study, and t h e corresponding m e d i a n day of t h e 10 largest and smallest T m a x and T m i n ranges. TABLE (0.6745 a ) m a x range (0.6745 a ) m a x range m e d i a n date m i n range m i n range m e d i a n date M a x range/ min range Asheville 30 Jan 1.3 5 Aug 3.9 Charlotte 4 Feb 1.4 15 Aug 3.7 Fayetteville 5.5 18 Feb 1.4 29 Jul 3.9 3.7 Station Greensboro 5.5 II Feb 1.5 Raleigh-Durham 5.5 18 Feb 1.5 14 Aug 3.7 Wilmington 26 Jan 1.2 6 Aug 4.3 3.9 5.3 8 Feb 1.4 5 Aug (0.6745 <x) m a x range m a x range m e d i a n date m i n range m i n range m e d i a n date Asheville 4.9 II Jan 0.8 27 Jul 6.1 Charlotte 4.6 14 Jan 1 Aug Fayetteville 4.5 25 Dec 5.0 Greensboro 4.9 14 Jan 5.4 Raleigh-Durham 19 Jan 1.0 Wilmington 3 Jan 1.0 26 Jul Average 4.9 9 Jan 26 Jul 5.3 Average Station 7 7 2 I BATIS- JUNE 2006 M a x range/ m i n range

2. N C D C - n o r m a l i z e d T m a x and T m i n ( F ) for t h e first 10 days of January at Greensboro, along w i t h the upper and lower limits of T m a x and T m i n ranges, calculated by (0.6745 * standard deviation), and added to and subtracted f r o m t h e n o r m a l t e m p e r a t u r e. TABLE Day range Upper normal Lower range Upper normal Lower 1 Jan 13.5 54.0 47.3 40.6 11.5 28.8 23.0 2 Jan 14.8 54.6 47.1 39.8 11.6 28.8 23.0 3 Jan 13.6 54.0 47.1 40.4 11.7 28.7 22.8 4 Jan 11.3 52.8 47.2 41.6 12.0 28.6 22.6 5 Jan 15.7 5 47.2 39.3 13.0 35.0 28.5 22.0 22.1 6 Jan 12.2 53.3 47.2 41.1 12.8 34.9 28.5 7 Jan 13.1 53.5 47.0 40.5 12.0 34.3 28.3 22.3 8 Jan 10.7 52.2 46.9 41.6 II.1 33.9 28.3 22.7 9 Jan 13.3 53.6 47.0 40.4 10.5 33.5 28.3 23.1 55.3 47.1 36.0 28.4 20.8 10 Jan 16.4 38.9 represents, while the second represents. The numbers to the left of the thermometers are the actual and for the day, while the numbers on the right are the normal ranges and historical record high (low) () for that day. A basic animation was rendered for this graphic, where the "mercury" rises to the actual temperatures, whose numbers then appear. Then, the numbers for the normal ranges, and finally the historical extremes, appear. A script could be written to make this graphic easy to utilize day to day make the mercury rise to a particular location, and place the normal ranges and extremes in appropriate relative locations along the mercury tube. C O N C L U S I O N S. With environmental issues often at the forefront of global politics, a need always exists for those involved with meteorology, climatology, and environmentalism to develop and utilize the most accurate data possible in the most accurate way possible. When comparing temperature records, three incorporating factors generally exist: what the temperature is, what it has been, and, if pertinent, what it will be. The "has beens," or the past records, are how we gauge where we are today climatologically. The use of 30-yr normal temperatures as comparisons to current weekly (or monthly, seasonaly, etc.) temperatures is arguably a very valid technique. But, as a comparison to daily observed temperatures, which broadcast meteorologists often utilize, the normal temperatures can be misleading. This study shows, and casual experience undoubtedly suggests, that variance in day-to-day temperatures is often large, especially in cooler months. This variance is AMERICAN METEOROLOGICAL SOCIETY 15.3 FIG. 3. A n e x a m p l e of how t h e n o r m a l t e m p e r a t u r e ranges could be presented on-air. It is a n i m a t e d such t h a t the m e r c u r y rises to the actual t e m p e r a t u r e s (75 and 47 F), t h e n t h e n o r m a l ranges ( 7 l - 7 9, 4 4-5 0 F ) a p p e a r, and finally t h e r e c o r d t e m p e r a t u r e s ( 9 5 and 29 F) are shown. A good script would m a k e this graphic easy to utilize in day-to-day broadcasts. not shown with 30-yr daily normal temperatures, and to present this normal temperature as "where we usually are this time of year" or "where we should be" often gives the public the wrong impression of current weather as compared to past weather. Presenting the public with a 30-yr normal range of temperatures gives a more accurate and representative idea of what the temperatures usually are like at any particular time of the year. The results of this study show the expected increase variance in cooler months, and a higher variance with than for most of the year. However, with temperature ranges sometimes as high as 5.5 C (9.9 F), JUNE 2006 BAflS- I 7 7 3

and varying somewhat from one day to the next, the normal ranges may give the impression of inaccuracy. It is therefore important to ensure that the function of this range be made clear, or to use a smaller normal probability. A smaller probability would create a smaller temperature range and reduce the magnitude of daily variability in the ranges. This study uses 50%, as a reference value, but using a probability of less than 50% may significantly reduce the usability and clarity of this technique. A fine line is present here between the range's viability in the public, its statistical significance, and its scientific value. Other statistical techniques could be performed to come up with a normal range. Because of its simplicity, and because it is applied to six separate locations, the process presented here is easily applicable to any location with an uninterrupted, quality-controlled dataset and with a near-normal distribution of daily temperature variance. An informal survey was performed as part of Lupo et al. (2003), a study that also addresses the question of "what is normal?" When presented with that study's graphical representations of its calculations of temperature variability, 65.4% of the 292 people surveyed found the presentation to be easy or somewhat easy to understand. Thus, it is possible to present this type of product so that a majority of people understand it. Further tests could be performed to minimize the transition from the normal temperature currently utilized. A C K N O W L E D G M E N T S. This study was supported by the State Climate Office of North Carolina and WRALTV5. REFERENCES Bingham, C., 1961: Distributions of weekly averages of diurnal temperature means and ranges about harmonic curves. Mon. Wea. Rev., 89, 357-367. Bruhn, J. A., W. E. Fry, and G. W. Fick, 1980: Simulation of daily weather data using theoretical probability distributions. /. Appl. Meteor., 19, 1029-1036. Lupo, A. R., and Coauthors, 2003: The presentation of temperature information in television broadcasts: What is normal? Nat. Wea. Digest, 27, 53-58. NCDC, cited 2004: Daily station normals. Climatology of the U.S. No. 84. [Available online at www.ncdc. noaa.gov/.] Railroads and Weather From Fogs to Floods and Heat to Hurricanes, the Impacts of Weather and Climate on American Railroading More than 120 photographs arid charts Railroads The most damaging storms of the last century Railroad response to Katrina From Fog* to Floods and Heal to Hurricanes, t h e ^ ^ ^ J <>F Weather and Cfimate on American Railroading "Railroads and Weather is a must-read for meteorologists wanting to understand the impact of their products on the rail community...as well as railroad operations personnel, and history buffs who just love trains." Richard A. Wagoner, National Center for Atmospheric Research Stanley Changnon Order from the A M S Today By Stanley A. Changnon 2006 136pp, hardbound, AMS order code: Rirds $24.95 member, $29.95 non-member ISBN 10: 1-878220-73-X Call (617) 227-2426, ext. 209 or 258, or email amsorder@ametsoc.org. Send pre-paid orders to the American Meteorological Society: AMS Order Dept., 45 Beacon Street, Boston, ISBN 13: 978-1-878330-73-8 774 I BATIS- JUNE 2006 www.ametsoc.org M A 02108-3693